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name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Fu, L"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" 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is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Fu%2C+L&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Fu%2C+L&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Fu%2C+L&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.11215">arXiv:2411.11215</a> <span> [<a href="https://arxiv.org/pdf/2411.11215">pdf</a>, <a href="https://arxiv.org/ps/2411.11215">ps</a>, <a href="https://arxiv.org/format/2411.11215">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Hypergeometric $\ell$-adic sheaves for reductive groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lei Fu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xuanyou Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.11215v1-abstract-short" style="display: inline;"> We define the hypergeometric exponential sum associated to a finite family of representations of a reductive group over a finite field. We introduce the hypergeometric $\ell$-adic sheaf to describe the behavior of the hypergeometric exponential sum. It is a perverse sheaf, and it is the counterpart in characteristic $p$ of the $A$-hypergeometric $\mathcal D$-module introduced by Kapranov. Using th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11215v1-abstract-full').style.display = 'inline'; document.getElementById('2411.11215v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.11215v1-abstract-full" style="display: none;"> We define the hypergeometric exponential sum associated to a finite family of representations of a reductive group over a finite field. We introduce the hypergeometric $\ell$-adic sheaf to describe the behavior of the hypergeometric exponential sum. It is a perverse sheaf, and it is the counterpart in characteristic $p$ of the $A$-hypergeometric $\mathcal D$-module introduced by Kapranov. Using the theory of the Fourier transform for vector bundles over a general base developed by Wang, we are able to study the hypergeometric $\ell$-adic sheaf via the hypergeometric $\mathcal D$-module. We apply our results to the estimation of the hypergeometric exponential sum. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11215v1-abstract-full').style.display = 'none'; document.getElementById('2411.11215v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Secondary 14M27; 11L07. Primary 14F10; 14F20; Secondary 14M27; 11L07. Primary 14F10; 14F20; Secondary 14M27; 11L07 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.10647">arXiv:2411.10647</a> <span> [<a href="https://arxiv.org/pdf/2411.10647">pdf</a>, <a href="https://arxiv.org/ps/2411.10647">ps</a>, <a href="https://arxiv.org/format/2411.10647">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> </div> </div> <p class="title is-5 mathjax"> False Discovery Control in Multiple Testing: A Brief Overview of Theories and Methodologies </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=He%2C+J">Jianliang He</a>, <a href="/search/math?searchtype=author&query=Gang%2C+B">Bowen Gang</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Luella Fu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.10647v1-abstract-short" style="display: inline;"> As the volume and complexity of data continue to expand across various scientific disciplines, the need for robust methods to account for the multiplicity of comparisons has grown widespread. A popular measure of type 1 error rate in multiple testing literature is the false discovery rate (FDR). The FDR provides a powerful and practical approach to large-scale multiple testing and has been success… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.10647v1-abstract-full').style.display = 'inline'; document.getElementById('2411.10647v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.10647v1-abstract-full" style="display: none;"> As the volume and complexity of data continue to expand across various scientific disciplines, the need for robust methods to account for the multiplicity of comparisons has grown widespread. A popular measure of type 1 error rate in multiple testing literature is the false discovery rate (FDR). The FDR provides a powerful and practical approach to large-scale multiple testing and has been successfully used in a wide range of applications. The concept of FDR has gained wide acceptance in the statistical community and various methods has been proposed to control the FDR. In this work, we review the latest developments in FDR control methodologies. We also develop a conceptual framework to better describe this vast literature; understand its intuition and key ideas; and provide guidance for the researcher interested in both the application and development of the methodology. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.10647v1-abstract-full').style.display = 'none'; document.getElementById('2411.10647v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.05883">arXiv:2410.05883</a> <span> [<a href="https://arxiv.org/pdf/2410.05883">pdf</a>, <a href="https://arxiv.org/format/2410.05883">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Signal Processing">eess.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Improved PCRLB for radar tracking in clutter with geometry-dependent target measurement uncertainty and application to radar trajectory control </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Shi%2C+Y">Yifang Shi</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yu Zhang</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Linjiao Fu</a>, <a href="/search/math?searchtype=author&query=Peng%2C+D">Dongliang Peng</a>, <a href="/search/math?searchtype=author&query=Lu%2C+Q">Qiang Lu</a>, <a href="/search/math?searchtype=author&query=Choi%2C+J+W">Jee Woong Choi</a>, <a href="/search/math?searchtype=author&query=Farina%2C+A">Alfonso Farina</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.05883v1-abstract-short" style="display: inline;"> In realistic radar tracking, target measurement uncertainty (TMU) in terms of both detection probability and measurement error covariance is significantly affected by the target-to-radar (T2R) geometry. However, existing posterior Cramer-Rao Lower Bounds (PCRLBs) rarely investigate the fundamental impact of T2R geometry on target measurement uncertainty and eventually on mean square error (MSE) of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.05883v1-abstract-full').style.display = 'inline'; document.getElementById('2410.05883v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.05883v1-abstract-full" style="display: none;"> In realistic radar tracking, target measurement uncertainty (TMU) in terms of both detection probability and measurement error covariance is significantly affected by the target-to-radar (T2R) geometry. However, existing posterior Cramer-Rao Lower Bounds (PCRLBs) rarely investigate the fundamental impact of T2R geometry on target measurement uncertainty and eventually on mean square error (MSE) of state estimate, inevitably resulting in over-conservative lower bound. To address this issue, this paper firstly derives the generalized model of target measurement error covariance for bistatic radar with moving receiver and transmitter illuminating any type of signal, along with its approximated solution to specify the impact of T2R geometry on error covariance. Based upon formulated TMU model, an improved PCRLB (IPCRLB) fully accounting for both measurement origin uncertainty and geometry-dependent TMU is then re-derived, both detection probability and measurement error covariance are treated as state-dependent parameters when differentiating log-likelihood with respect to target state. Compared to existing PCRLBs that partially or completely ignore the dependence of target measurement uncertainty on T2R geometry, proposed IPCRLB provides a much accurate (less-conservative) lower bound for radar tracking in clutter with geometry-dependent TMU. The new bound is then applied to radar trajectory control to effectively optimize T2R geometry and exhibits least uncertainty of acquired target measurement and more accurate state estimate for bistatic radar tracking in clutter, compared to state-of-the-art trajectory control methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.05883v1-abstract-full').style.display = 'none'; document.getElementById('2410.05883v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages,12 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> F.2.1 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.03599">arXiv:2407.03599</a> <span> [<a href="https://arxiv.org/pdf/2407.03599">pdf</a>, <a href="https://arxiv.org/ps/2407.03599">ps</a>, <a href="https://arxiv.org/format/2407.03599">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> The exotic inverted Kloosterman sum </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lei Fu</a>, <a href="/search/math?searchtype=author&query=Wan%2C+D">Daqing Wan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.03599v1-abstract-short" style="display: inline;"> Let $B$ be a product of finitely many finite fields containing $\mathbb F_q$, $蠄:\mathbb F_q\to \overline{\mathbb Q}_\ell^*$ a nontrivial additive character, and $蠂: B^*\to \overline{\mathbb Q}_\ell^*$ a multiplicative character. Katz introduced the so-called exotic inverted Kloosterman sum \begin{eqnarray*} \mathrm{EIK}(\mathbb F_q, a):=\sum_{\substack{x\in B^* \\ \mathrm{Tr}_{B/\mathbb F_q}(x)\n… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.03599v1-abstract-full').style.display = 'inline'; document.getElementById('2407.03599v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.03599v1-abstract-full" style="display: none;"> Let $B$ be a product of finitely many finite fields containing $\mathbb F_q$, $蠄:\mathbb F_q\to \overline{\mathbb Q}_\ell^*$ a nontrivial additive character, and $蠂: B^*\to \overline{\mathbb Q}_\ell^*$ a multiplicative character. Katz introduced the so-called exotic inverted Kloosterman sum \begin{eqnarray*} \mathrm{EIK}(\mathbb F_q, a):=\sum_{\substack{x\in B^* \\ \mathrm{Tr}_{B/\mathbb F_q}(x)\not =0\\ \mathrm{N}_{B/\mathbb F_q}(x)=a}} 蠂(x)蠄\Big(\frac{1}{\mathrm{Tr}_{B/\mathbb F_q}(x)}\Big), \ \ a\in \mathbb F_q^*. \end{eqnarray*} We estimate this sum using $\ell$-adic cohomology theory. Our main result is that, up to a trivial term, the associated exotic inverted Kloosterman sheaf is lisse of rank at most $2(n+1)$ and mixed of weight at most $n$, where $n+1 = \dim_{\mathbb F_q}B$. Up to a trivial main term, this gives the expected square root cancellation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.03599v1-abstract-full').style.display = 'none'; document.getElementById('2407.03599v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14F20; 11L05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.10106">arXiv:2406.10106</a> <span> [<a href="https://arxiv.org/pdf/2406.10106">pdf</a>, <a href="https://arxiv.org/ps/2406.10106">ps</a>, <a href="https://arxiv.org/format/2406.10106">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Equidistribution of Kloosterman sums over function fields </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lei Fu</a>, <a href="/search/math?searchtype=author&query=Lau%2C+Y">Yuk-Kam Lau</a>, <a href="/search/math?searchtype=author&query=Li%2C+W+W">Wen-Ching Winnie Li</a>, <a href="/search/math?searchtype=author&query=Xi%2C+P">Ping Xi</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.10106v1-abstract-short" style="display: inline;"> We prove the Sato--Tate distribution of Kloosterman sums over function fields with explicit error terms, when the places vary in arithmetic progressions or short intervals. A joint Sato--Tate distribution of two ``different" exponential sums is also proved. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.10106v1-abstract-full" style="display: none;"> We prove the Sato--Tate distribution of Kloosterman sums over function fields with explicit error terms, when the places vary in arithmetic progressions or short intervals. A joint Sato--Tate distribution of two ``different" exponential sums is also proved. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.10106v1-abstract-full').style.display = 'none'; document.getElementById('2406.10106v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.09633">arXiv:2406.09633</a> <span> [<a href="https://arxiv.org/pdf/2406.09633">pdf</a>, <a href="https://arxiv.org/ps/2406.09633">ps</a>, <a href="https://arxiv.org/format/2406.09633">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> An effective Deligne's equidistribution theorem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lei Fu</a>, <a href="/search/math?searchtype=author&query=Lau%2C+Y">Yuk-Kam Lau</a>, <a href="/search/math?searchtype=author&query=Xi%2C+P">Ping Xi</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.09633v2-abstract-short" style="display: inline;"> We prove an Erd艖s--Tur谩n type inequality for compact Lie groups, from which we deduce an effective version of Deligne's equidistribution theorem. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.09633v2-abstract-full" style="display: none;"> We prove an Erd艖s--Tur谩n type inequality for compact Lie groups, from which we deduce an effective version of Deligne's equidistribution theorem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.09633v2-abstract-full').style.display = 'none'; document.getElementById('2406.09633v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Improve the estimates in the previous version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11K38; 22E46; 14F20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.06244">arXiv:2309.06244</a> <span> [<a href="https://arxiv.org/pdf/2309.06244">pdf</a>, <a href="https://arxiv.org/format/2309.06244">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Hochschild cohomology of Hilbert schemes of points on surfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Belmans%2C+P">Pieter Belmans</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Krug%2C+A">Andreas Krug</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2309.06244v2-abstract-short" style="display: inline;"> We compute the Hochschild cohomology of Hilbert schemes of points on surfaces and observe that it is, in general, not determined solely by the Hochschild cohomology of the surface, but by its "Hochschild-Serre cohomology": the bigraded vector space obtained by taking Hochschild homologies with coefficients in powers of the Serre functor. As applications, we obtain various consequences on the defor… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.06244v2-abstract-full').style.display = 'inline'; document.getElementById('2309.06244v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.06244v2-abstract-full" style="display: none;"> We compute the Hochschild cohomology of Hilbert schemes of points on surfaces and observe that it is, in general, not determined solely by the Hochschild cohomology of the surface, but by its "Hochschild-Serre cohomology": the bigraded vector space obtained by taking Hochschild homologies with coefficients in powers of the Serre functor. As applications, we obtain various consequences on the deformation theory of the Hilbert schemes; in particular, we recover and extend results of Fantechi, Boissi猫re, and Hitchin. Our method is to compute more generally for any smooth proper algebraic variety $X$ the Hochschild-Serre cohomology of the symmetric quotient stack $[X^n/\mathfrak{S}_n]$, in terms of the Hochschild-Serre cohomology of $X$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.06244v2-abstract-full').style.display = 'none'; document.getElementById('2309.06244v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">45 pages, added further evidence for conjecture</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.03368">arXiv:2303.03368</a> <span> [<a href="https://arxiv.org/pdf/2303.03368">pdf</a>, <a href="https://arxiv.org/ps/2303.03368">ps</a>, <a href="https://arxiv.org/format/2303.03368">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Maximal real varieties from moduli constructions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2303.03368v2-abstract-short" style="display: inline;"> For a complex manifold equipped with an anti-holomorphic involution, which is referred to as a real variety, the Smith-Thom inequality states that the total $\mathbb{F}_2$-Betti number of the real locus is not greater than the total $\mathbb{F}_2$-Betti number of the ambient complex manifold. A real variety is called maximal if the equality holds. In this paper, we present a series of new construc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.03368v2-abstract-full').style.display = 'inline'; document.getElementById('2303.03368v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.03368v2-abstract-full" style="display: none;"> For a complex manifold equipped with an anti-holomorphic involution, which is referred to as a real variety, the Smith-Thom inequality states that the total $\mathbb{F}_2$-Betti number of the real locus is not greater than the total $\mathbb{F}_2$-Betti number of the ambient complex manifold. A real variety is called maximal if the equality holds. In this paper, we present a series of new constructions of maximal real varieties by exploring moduli spaces of certain objects on a maximal real variety. Our results establish the maximality of the following real varieties: - Moduli spaces of stable vector bundles of coprime rank and degree over a maximal smooth projective real curve (known as Brugall茅-Schaffhauser's theorem, with a short new proof presented in this work); the same result holds for moduli spaces of stable parabolic vector bundles. - Moduli spaces of stable Higgs bundles of coprime rank and degree over a maximal smooth projective real curve, providing maximal hyper-K盲hler examples. - If a real variety has non-empty real locus and maximal Hilbert square, then the variety itself and its Hilbert cube are maximal. This is always the case for maximal real smooth cubic threefolds, but never the case for maximal real smooth cubic fourfolds. - Punctual Hilbert schemes on a maximal real projective surface with vanishing first $\mathbb{F}_2$-Betti number and connected real locus, such as $\mathbb{R}$-rational maximal real surfaces and some generalized Dolgachev surfaces. - Moduli spaces of stable sheaves on the real projective plane, or more generally, on an $\mathbb{R}$-rational maximal Poisson surface. We also observe that maximality is a motivic property when interpreted as equivariant formality. Furthermore, any smooth projective real variety motivated by maximal ones is also maximal. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.03368v2-abstract-full').style.display = 'none'; document.getElementById('2303.03368v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 April, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">V2: various improvements. Comments welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14P25; 14H60; 14J60; 14C05; 14C15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2207.01122">arXiv:2207.01122</a> <span> [<a href="https://arxiv.org/pdf/2207.01122">pdf</a>, <a href="https://arxiv.org/ps/2207.01122">ps</a>, <a href="https://arxiv.org/format/2207.01122">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1090/jag/836">10.1090/jag/836 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The Tate Conjecture for even dimensional Gushel-Mukai varieties in characteristic $p\geq 5$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Moonen%2C+B">Ben Moonen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2207.01122v2-abstract-short" style="display: inline;"> We study Gushel-Mukai (GM) varieties of dimension 4 or 6 in characteristic $p$. Our main result is the Tate conjecture for all such varieties over finitely generated fields of characteristic $p\geq 5$. In the case of GM sixfolds, we follow the method used by Madapusi Pera in his proof of the Tate conjecture for K3 surfaces. As input for this, we prove a number of basic results about GM sixfolds, s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.01122v2-abstract-full').style.display = 'inline'; document.getElementById('2207.01122v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2207.01122v2-abstract-full" style="display: none;"> We study Gushel-Mukai (GM) varieties of dimension 4 or 6 in characteristic $p$. Our main result is the Tate conjecture for all such varieties over finitely generated fields of characteristic $p\geq 5$. In the case of GM sixfolds, we follow the method used by Madapusi Pera in his proof of the Tate conjecture for K3 surfaces. As input for this, we prove a number of basic results about GM sixfolds, such as the fact that there are no nonzero global vector fields. For GM fourfolds, we prove the Tate conjecture by reducing it to the case of GM sixfolds by making use of the notion of generalised partners plus the fact that generalised partners in characteristic 0 have isomorphic Chow motives in the middle degree. Several steps in the proofs rely on results in characteristic 0 that are proven our paper "Algebraic cycles on Gushel-Mukai varieties", 脡pijournal G茅om茅trie Alg茅brique, Volume sp茅cial en l'honneur de Claire Voisin, 2024. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.01122v2-abstract-full').style.display = 'none'; document.getElementById('2207.01122v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 July, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">64 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14G17; 14C25; 14J20; 14G35; 14J45 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Algebraic Geometry, electronically published on November 15, 2024 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2207.01118">arXiv:2207.01118</a> <span> [<a href="https://arxiv.org/pdf/2207.01118">pdf</a>, <a href="https://arxiv.org/format/2207.01118">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.46298/epiga.2024.9815">10.46298/epiga.2024.9815 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Algebraic cycles on Gushel-Mukai varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Moonen%2C+B">Ben Moonen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2207.01118v3-abstract-short" style="display: inline;"> We study algebraic cycles on complex Gushel-Mukai (GM) varieties. We prove the generalised Hodge conjecture, the (motivated) Mumford-Tate conjecture, and the generalised Tate conjecture for all GM varieties. We compute all integral Chow groups of GM varieties, except for the only two infinite-dimensional cases (1-cycles on GM fourfolds and 2-cycles on GM sixfolds). We prove that if two GM varietie… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.01118v3-abstract-full').style.display = 'inline'; document.getElementById('2207.01118v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2207.01118v3-abstract-full" style="display: none;"> We study algebraic cycles on complex Gushel-Mukai (GM) varieties. We prove the generalised Hodge conjecture, the (motivated) Mumford-Tate conjecture, and the generalised Tate conjecture for all GM varieties. We compute all integral Chow groups of GM varieties, except for the only two infinite-dimensional cases (1-cycles on GM fourfolds and 2-cycles on GM sixfolds). We prove that if two GM varieties are generalised partners or generalised duals, their rational Chow motives in middle degree are isomorphic. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.01118v3-abstract-full').style.display = 'none'; document.getElementById('2207.01118v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 July, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages, final version, in special volume in honour of C. Voisin</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14C15; 14C25; 14C30; 14J45; 14F08 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> 脙聣pijournal de G脙漏om脙漏trie Alg脙漏brique, Special volume in honour of Claire Voisin (July 2, 2024) epiga:9815 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2203.10391">arXiv:2203.10391</a> <span> [<a href="https://arxiv.org/pdf/2203.10391">pdf</a>, <a href="https://arxiv.org/ps/2203.10391">ps</a>, <a href="https://arxiv.org/format/2203.10391">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Unpolarized Shafarevich conjectures for hyper-K盲hler varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Li%2C+Z">Zhiyuan Li</a>, <a href="/search/math?searchtype=author&query=Takamatsu%2C+T">Teppei Takamatsu</a>, <a href="/search/math?searchtype=author&query=Zou%2C+H">Haitao Zou</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2203.10391v1-abstract-short" style="display: inline;"> The Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was proved by Y. She. For hyper-K盲hler varieties, which are higher-dimensional analogs of K3 surfaces, Y. Andr茅 proved the Shafarevich conjecture for hyper-K盲hler v… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.10391v1-abstract-full').style.display = 'inline'; document.getElementById('2203.10391v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2203.10391v1-abstract-full" style="display: none;"> The Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was proved by Y. She. For hyper-K盲hler varieties, which are higher-dimensional analogs of K3 surfaces, Y. Andr茅 proved the Shafarevich conjecture for hyper-K盲hler varieties of a given dimension and admitting a very ample polarization of bounded degree. In this paper, we provide a unification of both results by proving the (unpolarized) Shafarevich conjecture for hyper-K盲hler varieties in a given deformation type. We also discuss the cohomological generalization of the Shafarevich conjecture by replacing the good reduction condition by the unramifiedness of the cohomology, where our results are subject to a certain necessary assumption on the faithfulness of the action of the automorphism group on cohomology. In a similar fashion, generalizing a result of Orr and Skorobogatov on K3 surfaces, we prove the finiteness of geometric isomorphism classes of hyper-K盲hler varieties of CM type in a given deformation type defined over a number field with bounded degree. A key to our approach to these results is a uniform Kuga--Satake map, inspired by She's work, and we study its arithmetic properties, which are of independent interest. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.10391v1-abstract-full').style.display = 'none'; document.getElementById('2203.10391v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">42 pages. Comments welcome. This paper supersedes arXiv:2201.03350 and arXiv:2201.00482</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14G35; 14J28; 14J42; 11G15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2203.03632">arXiv:2203.03632</a> <span> [<a href="https://arxiv.org/pdf/2203.03632">pdf</a>, <a href="https://arxiv.org/ps/2203.03632">ps</a>, <a href="https://arxiv.org/format/2203.03632">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> On the stability of orthogonally Jensen additive and quadratic functional equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Linlin Fu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Q">Qi Liu</a>, <a href="/search/math?searchtype=author&query=Li%2C+Y">Yongjin Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2203.03632v1-abstract-short" style="display: inline;"> We consider the stability of the orthogonal Jensen additive and quadratic equations in $F$-spaces, through applying and extending the approach to the proof of a 2010 result of W.Frchner and J.Sikorska, we presenting a new method to get the stability. Moreover, we work in a more general and natural condition than considered before by other antuors. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2203.03632v1-abstract-full" style="display: none;"> We consider the stability of the orthogonal Jensen additive and quadratic equations in $F$-spaces, through applying and extending the approach to the proof of a 2010 result of W.Frchner and J.Sikorska, we presenting a new method to get the stability. Moreover, we work in a more general and natural condition than considered before by other antuors. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.03632v1-abstract-full').style.display = 'none'; document.getElementById('2203.03632v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2203.03114">arXiv:2203.03114</a> <span> [<a href="https://arxiv.org/pdf/2203.03114">pdf</a>, <a href="https://arxiv.org/ps/2203.03114">ps</a>, <a href="https://arxiv.org/format/2203.03114">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> Ulam stability of an additive-quadratic functional equation in F-space and quasi-Banach spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Linlin Fu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Q">Qi Liu</a>, <a href="/search/math?searchtype=author&query=Li%2C+Y">Yongjin Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2203.03114v1-abstract-short" style="display: inline;"> By adopting the direct method and fixed point method, we prove that the Hyers-Ulam stability of the following additive-quadratic functional equation \begin{equation} f(x+y, z+w)+f(x-y, z-w)-2 f(x, z)-2 f(x, w)=0 \end{equation} in $尾$-homogeneous $F$-spaces and quasi-Banach spaces. There are some differences that we consider the target space with the $尾$-homogeneous norm and quasi-norm. Overcomin… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.03114v1-abstract-full').style.display = 'inline'; document.getElementById('2203.03114v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2203.03114v1-abstract-full" style="display: none;"> By adopting the direct method and fixed point method, we prove that the Hyers-Ulam stability of the following additive-quadratic functional equation \begin{equation} f(x+y, z+w)+f(x-y, z-w)-2 f(x, z)-2 f(x, w)=0 \end{equation} in $尾$-homogeneous $F$-spaces and quasi-Banach spaces. There are some differences that we consider the target space with the $尾$-homogeneous norm and quasi-norm. Overcoming the $尾$-homogeneous norm and quasi-norm bottlenecks, we get some new results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.03114v1-abstract-full').style.display = 'none'; document.getElementById('2203.03114v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: text overlap with arXiv:0905.2173 by other authors</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2203.03101">arXiv:2203.03101</a> <span> [<a href="https://arxiv.org/pdf/2203.03101">pdf</a>, <a href="https://arxiv.org/ps/2203.03101">ps</a>, <a href="https://arxiv.org/format/2203.03101">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> On the stability of orthogonal additivity in $尾$-homogeneous $F$-spaces and quasi-Banach spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+Q">Qi Liu</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Linlin Fu</a>, <a href="/search/math?searchtype=author&query=Li%2C+Y">Yongjin Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2203.03101v1-abstract-short" style="display: inline;"> In this paper, we study the stability of the orthogonal equation,which is closely related to the results by Wlodzimierz Fechner and Justyna Sikorska in 2010. There are some differences that we consider the target space with the \b{eta}-homogeneous norm and quasi-norm. Overcoming the \b{eta}-homogeneous norm and quasi-norm bottlenecks, we get some new results. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2203.03101v1-abstract-full" style="display: none;"> In this paper, we study the stability of the orthogonal equation,which is closely related to the results by Wlodzimierz Fechner and Justyna Sikorska in 2010. There are some differences that we consider the target space with the \b{eta}-homogeneous norm and quasi-norm. Overcoming the \b{eta}-homogeneous norm and quasi-norm bottlenecks, we get some new results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.03101v1-abstract-full').style.display = 'none'; document.getElementById('2203.03101v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2203.02353">arXiv:2203.02353</a> <span> [<a href="https://arxiv.org/pdf/2203.02353">pdf</a>, <a href="https://arxiv.org/ps/2203.02353">ps</a>, <a href="https://arxiv.org/format/2203.02353">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> </div> </div> <p class="title is-5 mathjax"> A remark on the higher torsion invariants for flat vector bundles with finite holonomy </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yeping Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2203.02353v2-abstract-short" style="display: inline;"> We show that the Igusa-Klein topological torsion and the Bismut-Lott analytic torsion are equivalent for any flat vector bundle whose holonomy is a finite subgroup of $\mathrm{GL}_n(\mathbb{Q})$. Our proof uses Artin's induction theorem in representation theory to reduce the problem to the special case of trivial flat line bundles, which is a recent result of Puchol, Zhu and the second author. The… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.02353v2-abstract-full').style.display = 'inline'; document.getElementById('2203.02353v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2203.02353v2-abstract-full" style="display: none;"> We show that the Igusa-Klein topological torsion and the Bismut-Lott analytic torsion are equivalent for any flat vector bundle whose holonomy is a finite subgroup of $\mathrm{GL}_n(\mathbb{Q})$. Our proof uses Artin's induction theorem in representation theory to reduce the problem to the special case of trivial flat line bundles, which is a recent result of Puchol, Zhu and the second author. The idea of using Artin's induction theorem appeared in a paper of Ohrt on the same topic, of which our present work is an improvement. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.02353v2-abstract-full').style.display = 'none'; document.getElementById('2203.02353v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">revised version, 8 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 58J52; 57Q10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2201.03350">arXiv:2201.03350</a> <span>  </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Unpolarized Shafarevich conjectures for hyper-K盲hler varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Li%2C+Z">Zhiyuan Li</a>, <a href="/search/math?searchtype=author&query=Zou%2C+H">Haitao Zou</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2201.03350v3-abstract-short" style="display: inline;"> Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was proved by Y. She. For hyper-K盲hler varieties, which are higher-dimensional analogs of K3 surfaces, Y. Andr茅 has verified the Shafarevich conjecture for hyper-K盲hler… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.03350v3-abstract-full').style.display = 'inline'; document.getElementById('2201.03350v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2201.03350v3-abstract-full" style="display: none;"> Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was proved by Y. She. For hyper-K盲hler varieties, which are higher-dimensional analogs of K3 surfaces, Y. Andr茅 has verified the Shafarevich conjecture for hyper-K盲hler varieties of a given dimension and admitting a very ample polarization of bounded degree. In this paper, we provide a unification of both results by proving the (unpolarized) Shafarevich conjecture for hyper-K盲hler varieties in a given deformation type. In a similar fashion, generalizing a result of Orr and Skorobogatov on K3 surfaces, we prove the finiteness of geometric isomorphism classes of hyper-K盲hler varieties of CM type in a given deformation type defined over a number field with bounded degree. A key to our approach is a uniform Kuga--Satake map, inspired by She's work, and we study its arithmetic properties, which are of independent interest. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.03350v3-abstract-full').style.display = 'none'; document.getElementById('2201.03350v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This preprint is superseded by arXiv:2203.10391</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14G35; 14J28; 14J42; 11G15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2112.02437">arXiv:2112.02437</a> <span> [<a href="https://arxiv.org/pdf/2112.02437">pdf</a>, <a href="https://arxiv.org/ps/2112.02437">ps</a>, <a href="https://arxiv.org/format/2112.02437">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Special cubic fourfolds, K3 surfaces and the Franchetta property </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Laterveer%2C+R">Robert Laterveer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2112.02437v1-abstract-short" style="display: inline;"> O'Grady conjectured that the Chow group of 0-cycles of the generic fiber of the universal family over the moduli space of polarized K3 surfaces of genus g is cyclic. This so-called generalized Franchetta conjecture has been solved only for low genera where there is a Mukai model (precisely, when g<11 and g=12, 13, 16, 18, 20), by the work of Pavic--Shen--Yin. In this paper, as a non-commutative an… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2112.02437v1-abstract-full').style.display = 'inline'; document.getElementById('2112.02437v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2112.02437v1-abstract-full" style="display: none;"> O'Grady conjectured that the Chow group of 0-cycles of the generic fiber of the universal family over the moduli space of polarized K3 surfaces of genus g is cyclic. This so-called generalized Franchetta conjecture has been solved only for low genera where there is a Mukai model (precisely, when g<11 and g=12, 13, 16, 18, 20), by the work of Pavic--Shen--Yin. In this paper, as a non-commutative analog, we study the Franchetta property for families of special cubic fourfolds (in the sense of Hassett) and relate it to O'Grady's conjecture for K3 surfaces. Most notably, by using special cubic fourfolds of discriminant 26, we prove O'Grady's generalized Franchetta conjecture for g=14, providing the first evidence beyond Mukai models. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2112.02437v1-abstract-full').style.display = 'none'; document.getElementById('2112.02437v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 December, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages. Comments are very welcome</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14C15; 14C25; 14C30; 14J28; 14J70; 14J45 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2109.14340">arXiv:2109.14340</a> <span> [<a href="https://arxiv.org/pdf/2109.14340">pdf</a>, <a href="https://arxiv.org/format/2109.14340">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Fluid Dynamics">physics.flu-dyn</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.cma.2021.114193">10.1016/j.cma.2021.114193 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Very-high-order TENO schemes with adaptive accuracy order and adaptive dissipation control </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lin Fu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2109.14340v1-abstract-short" style="display: inline;"> In this paper, a new family of very-high-order TENO schemes with adaptive accuracy order and adaptive dissipation control (TENO-AA) is proposed. The new framework allows for constructing arbitrarily high-order TENO schemes in a unified paradigm and the yielded nonlinear schemes gradually reduce to low-order reconstructions by judging the smoothness with the ENO-like stencil selection strategy. In… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.14340v1-abstract-full').style.display = 'inline'; document.getElementById('2109.14340v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2109.14340v1-abstract-full" style="display: none;"> In this paper, a new family of very-high-order TENO schemes with adaptive accuracy order and adaptive dissipation control (TENO-AA) is proposed. The new framework allows for constructing arbitrarily high-order TENO schemes in a unified paradigm and the yielded nonlinear schemes gradually reduce to low-order reconstructions by judging the smoothness with the ENO-like stencil selection strategy. In order to control the nonlinear numerical dissipation adaptively, the flow scales are first measured by examining the first-order undivided difference and the cut-off constant $C_T$ in the TENO weighting strategy is adapted based on the corresponding measurement. With one set of optimal parameters, the newly proposed TENO schemes are designed to deliver excellent performance for predicting highly compressible flows with a wide range of Mach numbers. While the new very-high-order TENO schemes feature good robustness for conventional gas dynamics, the ENO-property is well preserved with the assistant of a positivity-preserving flux limiter for extreme simulations. Without loss of generality, the typical eight- and ten-point TENO-AA schemes are constructed. A set of benchmark simulations are computed to demonstrate the performance of the proposed TENO schemes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.14340v1-abstract-full').style.display = 'none'; document.getElementById('2109.14340v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 September, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">33 pages, 15 figures, accepted in Computer Methods in Applied Mechanics and Engineering</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 76F40; 35L03; 35L65; 35D30; 35D40 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> G.1.8; G.1.4; G.1.1 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2105.02127">arXiv:2105.02127</a> <span> [<a href="https://arxiv.org/pdf/2105.02127">pdf</a>, <a href="https://arxiv.org/format/2105.02127">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computational Physics">physics.comp-ph</span> </div> </div> <p class="title is-5 mathjax"> A class of new high-order finite-volume TENO schemes for hyperbolic conservation laws with unstructured meshes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ji%2C+Z">Zhe Ji</a>, <a href="/search/math?searchtype=author&query=Liang%2C+T">Tian Liang</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Lin Fu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2105.02127v2-abstract-short" style="display: inline;"> The recently proposed high-order TENO scheme [Fu et al., Journal of Computational Physics, 305, pp.333-359] has shown great potential in predicting complex fluids owing to the novel weighting strategy, which ensures the high-order accuracy, the low numerical dissipation, and the sharp shock-capturing capability. However, the applications are still restricted to simple geometries with Cartesian or… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.02127v2-abstract-full').style.display = 'inline'; document.getElementById('2105.02127v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2105.02127v2-abstract-full" style="display: none;"> The recently proposed high-order TENO scheme [Fu et al., Journal of Computational Physics, 305, pp.333-359] has shown great potential in predicting complex fluids owing to the novel weighting strategy, which ensures the high-order accuracy, the low numerical dissipation, and the sharp shock-capturing capability. However, the applications are still restricted to simple geometries with Cartesian or curvilinear meshes. In this work, a new class of high-order shock-capturing TENO schemes for unstructured meshes are proposed. Similar to the standard TENO schemes and some variants of WENO schemes, the candidate stencils include one large stencil and several small third-order stencils. Following a strong scale-separation procedure, a tailored novel ENO-like stencil selection strategy is proposed such that the high-order accuracy is restored in smooth regions by selecting the candidate reconstruction on the large stencil while the ENO property is enforced near discontinuities by adopting the candidate reconstruction from smooth small stencils. The nonsmooth stencils containing genuine discontinuities are explicitly excluded from the final reconstruction, leading to excellent numerical stability. Different from the WENO concept, such unique sharp stencil selection retains the low numerical dissipation without sacrificing the shock-capturing capability. The newly proposed framework enables arbitrarily high-order TENO reconstructions on unstructured meshes. For conceptual verification, the TENO schemes with third- to sixth-order accuracy are constructed. Without parameter tuning case by case, the performance of the proposed TENO schemes is demonstrated by examining a set of benchmark cases with broadband flow length scales. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.02127v2-abstract-full').style.display = 'none'; document.getElementById('2105.02127v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">53 pages, 23 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35L20; 76J20; 65M06; 65M08; 65M22 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> I.1.2; G.1.8; F.2.1; G.1.1 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2104.12956">arXiv:2104.12956</a> <span> [<a href="https://arxiv.org/pdf/2104.12956">pdf</a>, <a href="https://arxiv.org/ps/2104.12956">ps</a>, <a href="https://arxiv.org/format/2104.12956">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> $\ell$-adic Tautological Systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lei Fu</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Lian%2C+B">Bong Lian</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+D">Dingxin Zhang</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+X">Xinwen Zhu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2104.12956v1-abstract-short" style="display: inline;"> Tautological systems was introduced in Lian-Yau as the system of differential equations satisfied by period integrals of hyperplane sections of some complex projective homogenous varieties. We introduce the $\ell$-adic tautological systems for the case where the ground field is of characteristic $p$. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2104.12956v1-abstract-full" style="display: none;"> Tautological systems was introduced in Lian-Yau as the system of differential equations satisfied by period integrals of hyperplane sections of some complex projective homogenous varieties. We introduce the $\ell$-adic tautological systems for the case where the ground field is of characteristic $p$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.12956v1-abstract-full').style.display = 'none'; document.getElementById('2104.12956v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 April, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14F20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2103.07728">arXiv:2103.07728</a> <span> [<a href="https://arxiv.org/pdf/2103.07728">pdf</a>, <a href="https://arxiv.org/ps/2103.07728">ps</a>, <a href="https://arxiv.org/format/2103.07728">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Stability manifolds of varieties with finite Albanese morphisms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Li%2C+C">Chunyi Li</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+X">Xiaolei Zhao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2103.07728v2-abstract-short" style="display: inline;"> For a smooth projective complex variety whose Albanese morphism is finite, we show that every Bridgeland stability condition on its bounded derived category of coherent sheaves is geometric, in the sense that all skyscraper sheaves are stable with the same phase. Furthermore, we describe the stability manifolds of irregular surfaces and abelian threefolds with Picard rank one, and show that they a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.07728v2-abstract-full').style.display = 'inline'; document.getElementById('2103.07728v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.07728v2-abstract-full" style="display: none;"> For a smooth projective complex variety whose Albanese morphism is finite, we show that every Bridgeland stability condition on its bounded derived category of coherent sheaves is geometric, in the sense that all skyscraper sheaves are stable with the same phase. Furthermore, we describe the stability manifolds of irregular surfaces and abelian threefolds with Picard rank one, and show that they are connected and contractible. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.07728v2-abstract-full').style.display = 'none'; document.getElementById('2103.07728v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version, to appear in Transactions of the AMS. Comments are still welcome</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14F08; 14K05; 14J60; 18G80 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2102.07546">arXiv:2102.07546</a> <span> [<a href="https://arxiv.org/pdf/2102.07546">pdf</a>, <a href="https://arxiv.org/ps/2102.07546">ps</a>, <a href="https://arxiv.org/format/2102.07546">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3934/era.2022004">10.3934/era.2022004 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Motives of moduli spaces of rank 3 vector bundles and Higgs bundles on a curve </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Hoskins%2C+V">Victoria Hoskins</a>, <a href="/search/math?searchtype=author&query=Lehalleur%2C+S+P">Simon Pepin Lehalleur</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2102.07546v2-abstract-short" style="display: inline;"> We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank 3 and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a completion of) the Grothendieck group of effective Chow motives to isomorphisms in the category of Chow motives. For the Higgs moduli space, we use motivic Bialy… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.07546v2-abstract-full').style.display = 'inline'; document.getElementById('2102.07546v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2102.07546v2-abstract-full" style="display: none;"> We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank 3 and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a completion of) the Grothendieck group of effective Chow motives to isomorphisms in the category of Chow motives. For the Higgs moduli space, we use motivic Bialynicki-Birula decompositions associated to a scaling action with variation of stability and wall-crossing for moduli spaces of rank 2 pairs, which occur in the fixed locus of this action. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.07546v2-abstract-full').style.display = 'none'; document.getElementById('2102.07546v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 December, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version appearing in a special issue "Birational geometry and moduli of projective varieties" of the Electronic Research Archive</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14H60; 14D20; 14C15; 14E05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Electron. res. arch. (2022) 30 (1), 66--89 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2011.14872">arXiv:2011.14872</a> <span> [<a href="https://arxiv.org/pdf/2011.14872">pdf</a>, <a href="https://arxiv.org/ps/2011.14872">ps</a>, <a href="https://arxiv.org/format/2011.14872">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Motives of moduli spaces of bundles on curves via variation of stability and flips </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Hoskins%2C+V">Victoria Hoskins</a>, <a href="/search/math?searchtype=author&query=Lehalleur%2C+S+P">Simon Pepin Lehalleur</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2011.14872v1-abstract-short" style="display: inline;"> We study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.14872v1-abstract-full').style.display = 'inline'; document.getElementById('2011.14872v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2011.14872v1-abstract-full" style="display: none;"> We study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall-crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall-crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulas for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.14872v1-abstract-full').style.display = 'none'; document.getElementById('2011.14872v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Comments welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14H60; 14D20; 14C15; 14E05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2010.13289">arXiv:2010.13289</a> <span> [<a href="https://arxiv.org/pdf/2010.13289">pdf</a>, <a href="https://arxiv.org/ps/2010.13289">ps</a>, <a href="https://arxiv.org/format/2010.13289">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jcp.2020.109960">10.1016/j.jcp.2020.109960 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A low-dissipation shock-capturing framework with flexible nonlinear dissipation control </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Y">Yue Li</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Lin Fu</a>, <a href="/search/math?searchtype=author&query=Adams%2C+N+A">Nikolaus A. Adams</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2010.13289v1-abstract-short" style="display: inline;"> In this work, a framework to construct arbitrarily high-order low-dissipation shock-capturing schemes with flexible and controllable nonlinear dissipation for convection-dominated problems is proposed. While a set of candidate stencils of incremental width is constructed, each one is indicated as smooth or nonsmooth by the ENO-like stencil selection procedure proposed in the targeted essentially n… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2010.13289v1-abstract-full').style.display = 'inline'; document.getElementById('2010.13289v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2010.13289v1-abstract-full" style="display: none;"> In this work, a framework to construct arbitrarily high-order low-dissipation shock-capturing schemes with flexible and controllable nonlinear dissipation for convection-dominated problems is proposed. While a set of candidate stencils of incremental width is constructed, each one is indicated as smooth or nonsmooth by the ENO-like stencil selection procedure proposed in the targeted essentially non-oscillatory (TENO) scheme [Fu et al., Journal of Computational Physics 305 (2016): 333-359]. Rather than being discarded directly as with TENO schemes, the nonsmooth candidates are filtered by an extra nonlinear limiter, such as a monotonicity-preserving (MP) limiter or a total variation diminishing (TVD) limiter. Consequently, high-order reconstruction is achieved by assembling candidate fluxes with optimal linear weights since they are either smooth reconstructions or filtered ones which feature good non-oscillation property. A weight renormalization procedure as with the standard TENO paradigm is not necessary. This new framework concatenates the concepts of TENO, WENO and other nonlinear limiters for shock-capturing, and provides a new insight to designing low-dissipation nonlinear schemes. Through the adaptation of nonlinear limiters, nonlinear dissipation in the newly proposed framework can be controlled separately without affecting the performance in smooth regions. Based on the proposed framework, a family of new six- and eight-point nonlinear schemes with controllable dissipation is proposed. A set of critical benchmark cases involving strong discontinuities and broadband fluctuations is simulated. Numerical results reveal that the proposed new schemes capture discontinuities sharply and resolve the high-wavenumber fluctuations with low dissipation, while maintaining the desired accuracy order in smooth regions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2010.13289v1-abstract-full').style.display = 'none'; document.getElementById('2010.13289v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 October, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">35 pages, 19 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 76N06; 76L05; 35L65; 35L50; 65M06; 65M08; </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2009.13173">arXiv:2009.13173</a> <span> [<a href="https://arxiv.org/pdf/2009.13173">pdf</a>, <a href="https://arxiv.org/ps/2009.13173">ps</a>, <a href="https://arxiv.org/format/2009.13173">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4171/DM/925">10.4171/DM/925 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Cubic fourfolds, Kuznetsov components and Chow motives </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Vial%2C+C">Charles Vial</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2009.13173v2-abstract-short" style="display: inline;"> We prove that the Chow motives of two smooth cubic fourfolds whose Kuznetsov components are Fourier-Mukai derived-equivalent are isomorphic as Frobenius algebra objects. As a corollary, we obtain that there exists a Galois-equivariant isomorphism between their l-adic cohomology Frobenius algebras. We also discuss the case where the Kuznetsov component of a smooth cubic fourfold is Fourier-Mukai de… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2009.13173v2-abstract-full').style.display = 'inline'; document.getElementById('2009.13173v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2009.13173v2-abstract-full" style="display: none;"> We prove that the Chow motives of two smooth cubic fourfolds whose Kuznetsov components are Fourier-Mukai derived-equivalent are isomorphic as Frobenius algebra objects. As a corollary, we obtain that there exists a Galois-equivariant isomorphism between their l-adic cohomology Frobenius algebras. We also discuss the case where the Kuznetsov component of a smooth cubic fourfold is Fourier-Mukai derived-equivalent to a K3 surface. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2009.13173v2-abstract-full').style.display = 'none'; document.getElementById('2009.13173v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 September, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Documenta Math. 28 (2023) 827-856 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2009.10959">arXiv:2009.10959</a> <span> [<a href="https://arxiv.org/pdf/2009.10959">pdf</a>, <a href="https://arxiv.org/ps/2009.10959">ps</a>, <a href="https://arxiv.org/format/2009.10959">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1093/imrn/rnaa349">10.1093/imrn/rnaa349 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Supersingular O'Grady varieties of dimension six </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Li%2C+Z">Zhiyuan Li</a>, <a href="/search/math?searchtype=author&query=Zou%2C+H">Haitao Zou</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2009.10959v2-abstract-short" style="display: inline;"> O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields of positive characteristic p greater than 2, called OG6 varieties. We show that a supersingular OG6 variety is unirational, its rational cohomology group is gen… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2009.10959v2-abstract-full').style.display = 'inline'; document.getElementById('2009.10959v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2009.10959v2-abstract-full" style="display: none;"> O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields of positive characteristic p greater than 2, called OG6 varieties. We show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin--Shioda conjecture, the supersingular Tate conjecture and the supersingular Bloch conjecture proposed in our previous work, in analogy with the theory of supersingular K3 surfaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2009.10959v2-abstract-full').style.display = 'none'; document.getElementById('2009.10959v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 September, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version. To appear in I.M.R.N</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14J28; 14J42; 14G17; 14D22; 14M20; 14C15; 14C25; </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2008.07438">arXiv:2008.07438</a> <span> [<a href="https://arxiv.org/pdf/2008.07438">pdf</a>, <a href="https://arxiv.org/ps/2008.07438">ps</a>, <a href="https://arxiv.org/format/2008.07438">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Networking and Internet Architecture">cs.NI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Systems and Control">eess.SY</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> </div> </div> <p class="title is-5 mathjax"> Analysis and Optimization for Large-Scale LoRa Networks: Throughput Fairness and Scalability </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lyu%2C+J">Jiangbin Lyu</a>, <a href="/search/math?searchtype=author&query=Yu%2C+D">Dan Yu</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Liqun Fu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2008.07438v3-abstract-short" style="display: inline;"> LoRa networks are pivotally enabling Long Range connectivity to low-cost and power-constrained user equipments (UEs) in a wide area, whereas a critical issue is to effectively allocate wireless resources to support potentially massive UEs while resolving the prominent near-far fairness issue, which is challenging due to the lack of tractable analytical model and the practical requirement for low-c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.07438v3-abstract-full').style.display = 'inline'; document.getElementById('2008.07438v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2008.07438v3-abstract-full" style="display: none;"> LoRa networks are pivotally enabling Long Range connectivity to low-cost and power-constrained user equipments (UEs) in a wide area, whereas a critical issue is to effectively allocate wireless resources to support potentially massive UEs while resolving the prominent near-far fairness issue, which is challenging due to the lack of tractable analytical model and the practical requirement for low-complexity and low-overhead design. Leveraging on stochastic geometry, especially the Poisson rain model, we derive (semi-) closed-form formulas for the aggregate interference distribution, packet success probability and hence system throughput in both single-cell and multi-cell setups with frequency reuse, by accounting for channel fading, random UE distribution, partial packet overlapping, and/or multi-gateway packet reception. The analytical formulas require only average channel statistics and spatial UE distribution, which enable tractable network performance evaluation and incubate our proposed Iterative Balancing (IB) method that quickly yields high-level policies of joint spreading factor (SF) allocation, power control, and duty cycle adjustment for gauging the average max-min UE throughput or supported UE density with rate requirements. Numerical results validate the analytical formulas and the effectiveness of our proposed optimization scheme, which greatly alleviates the near-far fairness issue and reduces the spatial power consumption, while significantly improving the cell-edge throughput as well as the spatial (sum) throughput for the majority of UEs, by adapting to the UE/gateway densities. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.07438v3-abstract-full').style.display = 'none'; document.getElementById('2008.07438v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 November, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">To appear in IEEE IOT Journal. Stochastic geometry-based framework to model/analyze large-scale LoRa networks with channel fading/aggregate interference/packet overlapping/multi-GW reception. Jointly optimize SF/Tx-power/duty-cycle based on channel statistics and UE distribution. Achieve both fairness/power savings and improve cell-edge throughput and spatial (sum) throughput for majority of UEs. arXiv admin note: text overlap with arXiv:1904.12300</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2007.04835">arXiv:2007.04835</a> <span> [<a href="https://arxiv.org/pdf/2007.04835">pdf</a>, <a href="https://arxiv.org/ps/2007.04835">ps</a>, <a href="https://arxiv.org/format/2007.04835">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00029-023-00832-3">10.1007/s00029-023-00832-3 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Motivic integration and the birational invariance of BCOV invariants </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yeping Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2007.04835v4-abstract-short" style="display: inline;"> Bershadsky, Cecotti, Ooguri, and Vafa constructed a real-valued invariant for Calabi--Yau manifolds, which is now called the BCOV torsion. Based on it, a metric-independent invariant, called the BCOV invariant, was constructed by Fang--Lu--Yoshikawa and Eriksson--Freixas i Montplet--Mourougane. The BCOV invariant is conjecturally related to the Gromov--Witten theory via mirror symmetry. Based upon… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.04835v4-abstract-full').style.display = 'inline'; document.getElementById('2007.04835v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2007.04835v4-abstract-full" style="display: none;"> Bershadsky, Cecotti, Ooguri, and Vafa constructed a real-valued invariant for Calabi--Yau manifolds, which is now called the BCOV torsion. Based on it, a metric-independent invariant, called the BCOV invariant, was constructed by Fang--Lu--Yoshikawa and Eriksson--Freixas i Montplet--Mourougane. The BCOV invariant is conjecturally related to the Gromov--Witten theory via mirror symmetry. Based upon the previous work of the second author, we prove the conjecture that birational Calabi--Yau manifolds have the same BCOV invariant. We also extend the construction of the BCOV invariant to Calabi--Yau varieties with Kawamata log terminal singularities and prove its birational invariance for Calabi--Yau varieties with canonical singularities. We provide an interpretation of our construction using the theory of motivic integration. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.04835v4-abstract-full').style.display = 'none'; document.getElementById('2007.04835v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 July, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version to appear in Selecta Mathematica</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14J32; 58J52; 14E18; 14J33; 14J17; 32Q25 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Sel. Math. New Ser. 29, 25 (2023) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.01371">arXiv:2006.01371</a> <span> [<a href="https://arxiv.org/pdf/2006.01371">pdf</a>, <a href="https://arxiv.org/ps/2006.01371">ps</a>, <a href="https://arxiv.org/format/2006.01371">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Signal Processing">eess.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Networking and Internet Architecture">cs.NI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Systems and Control">eess.SY</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1109/GLOBECOM42002.2020.9322588">10.1109/GLOBECOM42002.2020.9322588 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Energy-Efficient Cyclical Trajectory Design for UAV-Aided Maritime Data Collection in Wind </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yifan Zhang</a>, <a href="/search/math?searchtype=author&query=Lyu%2C+J">Jiangbin Lyu</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Liqun Fu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2006.01371v2-abstract-short" style="display: inline;"> Unmanned aerial vehicles (UAVs), especially fixed-wing ones that withstand strong winds, have great potential for oceanic exploration and research. This paper studies a UAV-aided maritime data collection system with a fixed-wing UAV dispatched to collect data from marine buoys. We aim to minimize the UAV's energy consumption in completing the task by jointly optimizing the communication time sched… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.01371v2-abstract-full').style.display = 'inline'; document.getElementById('2006.01371v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.01371v2-abstract-full" style="display: none;"> Unmanned aerial vehicles (UAVs), especially fixed-wing ones that withstand strong winds, have great potential for oceanic exploration and research. This paper studies a UAV-aided maritime data collection system with a fixed-wing UAV dispatched to collect data from marine buoys. We aim to minimize the UAV's energy consumption in completing the task by jointly optimizing the communication time scheduling among the buoys and the UAV's flight trajectory subject to wind effect, which is a non-convex problem and difficult to solve optimally. Existing techniques such as the successive convex approximation (SCA) method provide efficient sub-optimal solutions for collecting small/moderate data volume, whereas the solution heavily relies on the trajectory initialization and has not explicitly considered the wind effect, while the computational complexity and resulted trajectory complexity both become prohibitive for the task with large data volume. To this end, we propose a new cyclical trajectory design framework that can handle arbitrary data volume efficiently subject to wind effect. Specifically, the proposed UAV trajectory comprises multiple cyclical laps, each responsible for collecting only a subset of data and thereby significantly reducing the computational/trajectory complexity, which allows searching for better trajectory initialization that fits the buoys' topology and the wind. Numerical results show that the proposed cyclical scheme outperforms the benchmark one-flight-only scheme in general. Moreover, the optimized cyclical 8-shape trajectory can proactively exploit the wind and achieve lower energy consumption compared with the case without wind. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.01371v2-abstract-full').style.display = 'none'; document.getElementById('2006.01371v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 November, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Published in GLOBECOM2020. Investigated UAV-aided maritime data collection in wind, with joint trajectory and communications optimization for energy efficiency. Proposed new cyclical trajectory design that can handle arbitrary data volume with significantly reduced computational/trajectory complexity. Unveiled that the wind can be proactively utilized by our optimized trajectory</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> GLOBECOM 2020 - 2020 IEEE Global Communications Conference, 2020, pp. 1-6 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2003.14224">arXiv:2003.14224</a> <span> [<a href="https://arxiv.org/pdf/2003.14224">pdf</a>, <a href="https://arxiv.org/ps/2003.14224">ps</a>, <a href="https://arxiv.org/format/2003.14224">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.aim.2021.107655">10.1016/j.aim.2021.107655 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Categorical polynomial entropy </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fan%2C+Y">Yu-Wei Fan</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Ouchi%2C+G">Genki Ouchi</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2003.14224v2-abstract-short" style="display: inline;"> For classical dynamical systems, the polynomial entropy serves as a refined invariant of the topological entropy. In the setting of categorical dynamical systems, that is, triangulated categories endowed with an endofunctor, we develop the theory of categorical polynomial entropy, refining the categorical entropy defined by Dimitrov-Haiden-Katzarkov-Kontsevich. We justify this notion by showing th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2003.14224v2-abstract-full').style.display = 'inline'; document.getElementById('2003.14224v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2003.14224v2-abstract-full" style="display: none;"> For classical dynamical systems, the polynomial entropy serves as a refined invariant of the topological entropy. In the setting of categorical dynamical systems, that is, triangulated categories endowed with an endofunctor, we develop the theory of categorical polynomial entropy, refining the categorical entropy defined by Dimitrov-Haiden-Katzarkov-Kontsevich. We justify this notion by showing that for an automorphism of a smooth projective variety, the categorical polynomial entropy of the pullback functor on the derived category coincides with the polynomial growth rate of the induced action on cohomology. We also establish in general a Yomdin-type lower bound for the categorical polynomial entropy of an endofunctor in terms of the induced endomorphism on the numerical Grothendieck group of the category. As examples, we compute the categorical polynomial entropy for some standard functors like shifts, Serre functors, tensoring line bundles, automorphisms, spherical twists, P-twists, and so on, illustrating clearly how categorical polynomial entropy refines the study of categorical entropy and enables us to study the phenomenon of categorical trichotomy. A parallel theory of polynomial mass growth rate is developed in the presence of Bridgeland stability conditions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2003.14224v2-abstract-full').style.display = 'none'; document.getElementById('2003.14224v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 31 March, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version. Published in Advances in Mathematics</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> RIKEN-iTHEMS-Report-20 <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 18E30; 37B40; 14F05; 18E30; 16G20 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Advances in Mathematics 383 (2021) 107655 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2003.08796">arXiv:2003.08796</a> <span> [<a href="https://arxiv.org/pdf/2003.08796">pdf</a>, <a href="https://arxiv.org/ps/2003.08796">ps</a>, <a href="https://arxiv.org/format/2003.08796">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> On Katz's $(A,B)$-exponential sums </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lei Fu</a>, <a href="/search/math?searchtype=author&query=Wan%2C+D">Daqing Wan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2003.08796v1-abstract-short" style="display: inline;"> We deduce Katz's theorems for $(A,B)$-exponential sums over finite fields using $\ell$-adic cohomology and a theorem of Denef-Loeser, removing the hypothesis that $A+B$ is relatively prime to the characteristic $p$. In some degenerate cases, the Betti number estimate is improved using toric decomposition and Adolphson-Sperber's bound for the degree of $L$-functions. Applying the facial decompositi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2003.08796v1-abstract-full').style.display = 'inline'; document.getElementById('2003.08796v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2003.08796v1-abstract-full" style="display: none;"> We deduce Katz's theorems for $(A,B)$-exponential sums over finite fields using $\ell$-adic cohomology and a theorem of Denef-Loeser, removing the hypothesis that $A+B$ is relatively prime to the characteristic $p$. In some degenerate cases, the Betti number estimate is improved using toric decomposition and Adolphson-Sperber's bound for the degree of $L$-functions. Applying the facial decomposition theorem in \cite{W1}, we prove that the universal family of $(A,B)$-polynomials is generically ordinary for its $L$-function when $p$ is in certain arithmetic progression. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2003.08796v1-abstract-full').style.display = 'none'; document.getElementById('2003.08796v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 March, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14F20; 11T23 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2003.02407">arXiv:2003.02407</a> <span> [<a href="https://arxiv.org/pdf/2003.02407">pdf</a>, <a href="https://arxiv.org/format/2003.02407">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Atmospheric and Oceanic Physics">physics.ao-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Chaotic Dynamics">nlin.CD</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/5.0006626">10.1063/5.0006626 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The tipping times in an Arctic sea ice system under influence of extreme events </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+F">Fang Yang</a>, <a href="/search/math?searchtype=author&query=Zheng%2C+Y">Yayun Zheng</a>, <a href="/search/math?searchtype=author&query=Duan%2C+J">Jinqiao Duan</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Ling Fu</a>, <a href="/search/math?searchtype=author&query=Wiggins%2C+S">Stephen Wiggins</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2003.02407v2-abstract-short" style="display: inline;"> In light of the rapid recent retreat of Arctic sea ice, the extreme weather events triggering the variability in Arctic ice cover has drawn increasing attention. A non-Gaussian $伪$-stable L茅vy process is thought to be an appropriate model to describe such extreme event. The maximal likely trajectory, based on the nonlocal Fokker-Planck equation, is applied to a nonautonomous Arctic sea ice system… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2003.02407v2-abstract-full').style.display = 'inline'; document.getElementById('2003.02407v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2003.02407v2-abstract-full" style="display: none;"> In light of the rapid recent retreat of Arctic sea ice, the extreme weather events triggering the variability in Arctic ice cover has drawn increasing attention. A non-Gaussian $伪$-stable L茅vy process is thought to be an appropriate model to describe such extreme event. The maximal likely trajectory, based on the nonlocal Fokker-Planck equation, is applied to a nonautonomous Arctic sea ice system under $伪$-stable L茅vy noise. Two types of tipping times, the early-warning tipping time and the disaster-happening tipping time, are used to predict the critical time for the maximal likely transition from a perennially ice-covered state to a seasonally ice-free one, and from a seasonally ice-free state to a perennially ice-free one, respectively. We find that the increased intensity of extreme events results in shorter warning time for sea ice melting, and that an enhanced greenhouse effect will intensify this influence, making the arrival of warning time significantly earlier. Meanwhile, for the enhanced greenhouse effect, we discover that increased intensity and frequency of extreme events will advance the disaster-happening tipping time, in which an ice-free state is maintained throughout the year in the Arctic Ocean. Finally, we identify values of L茅vy index $伪$ and noise intensity $蔚$ in $伪蔚$-space that can trigger a transition between the Arctic sea ice state. These results provide an effective theoretical framework for studying Arctic sea ice variations under the influence of extreme events. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2003.02407v2-abstract-full').style.display = 'none'; document.getElementById('2003.02407v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 May, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 March, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2002.05490">arXiv:2002.05490</a> <span> [<a href="https://arxiv.org/pdf/2002.05490">pdf</a>, <a href="https://arxiv.org/ps/2002.05490">ps</a>, <a href="https://arxiv.org/format/2002.05490">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.5802/jep.166">10.5802/jep.166 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The generalized Franchetta conjecture for some hyper-K盲hler varieties, II </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Laterveer%2C+R">Robert Laterveer</a>, <a href="/search/math?searchtype=author&query=Vial%2C+C">Charles Vial</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2002.05490v2-abstract-short" style="display: inline;"> We prove the generalized Franchetta conjecture for the locally complete family of hyper-K盲hler eightfolds constructed by Lehn-Lehn-Sorger-van Straten (LLSS). As a corollary, we establish the Beauville-Voisin conjecture for very general LLSS eightfolds. The strategy consists in reducing to the Franchetta property for relative fourth powers of cubic fourfolds, by using the recent description of LLSS… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.05490v2-abstract-full').style.display = 'inline'; document.getElementById('2002.05490v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2002.05490v2-abstract-full" style="display: none;"> We prove the generalized Franchetta conjecture for the locally complete family of hyper-K盲hler eightfolds constructed by Lehn-Lehn-Sorger-van Straten (LLSS). As a corollary, we establish the Beauville-Voisin conjecture for very general LLSS eightfolds. The strategy consists in reducing to the Franchetta property for relative fourth powers of cubic fourfolds, by using the recent description of LLSS eightfolds as moduli spaces of semistable objects in the Kuznetsov component of the derived category of cubic fourfolds, together with its generalization to the relative setting due to Bayer-Lahoz-Macr矛-Nuer-Perry-Stellari. As a by-product, we compute the Chow motive of the Fano variety of lines on a smooth cubic hypersurface in terms of the Chow motive of the cubic hypersurface. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.05490v2-abstract-full').style.display = 'none'; document.getElementById('2002.05490v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages; improved version thanks to referees' comments</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14C25; 14C15; 14J42; 14J28; 14F08; 14J70; 14D20; 14H10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal de l'脡cole polytechnique -- Math茅matiques, Tome 8 (2021), pp. 1065--1097 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2002.04940">arXiv:2002.04940</a> <span> [<a href="https://arxiv.org/pdf/2002.04940">pdf</a>, <a href="https://arxiv.org/ps/2002.04940">ps</a>, <a href="https://arxiv.org/format/2002.04940">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1112/plms.12487">10.1112/plms.12487 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Derived categories of flips and cubic hypersurfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Belmans%2C+P">Pieter Belmans</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Raedschelders%2C+T">Theo Raedschelders</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2002.04940v2-abstract-short" style="display: inline;"> A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by describing the complement. As an application, we can lift the "quadratic Fano correspondence" (due to Galkin-Shinder) in the Grothendieck ring of varieties betw… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.04940v2-abstract-full').style.display = 'inline'; document.getElementById('2002.04940v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2002.04940v2-abstract-full" style="display: none;"> A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by describing the complement. As an application, we can lift the "quadratic Fano correspondence" (due to Galkin-Shinder) in the Grothendieck ring of varieties between a smooth cubic hypersurface, its Fano variety of lines, and its Hilbert square, to a semiorthogonal decomposition. We also show that the Hilbert square of a cubic hypersurface of dimension at least 3 is again a Fano variety, so in particular the Fano variety of lines on a cubic hypersurface is a Fano visitor. The most interesting case is that of a cubic fourfold, where this exhibits the first higher-dimensional hyperk盲hler variety as a Fano visitor. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.04940v2-abstract-full').style.display = 'none'; document.getElementById('2002.04940v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages, nearly identical to published version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Proceedings of the London Mathematical Society, 2022, Volume 125, Issue 6, Pages 1452-1482 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2001.06643">arXiv:2001.06643</a> <span> [<a href="https://arxiv.org/pdf/2001.06643">pdf</a>, <a href="https://arxiv.org/ps/2001.06643">ps</a>, <a href="https://arxiv.org/format/2001.06643">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00209-020-02682-7">10.1007/s00209-020-02682-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Menet%2C+G">Gr茅goire Menet</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2001.06643v2-abstract-short" style="display: inline;"> We extend a result of Guan by showing that the second Betti number of a 4-dimensional primitively symplectic orbifold is at most 23 and there are at most 91 singular points. The maximal possibility 23 can only occur in the smooth case. In addition to the known smooth examples with second Betti numbers 7 and 23, we provide examples of such orbifolds with second Betti numbers 3, 5, 6, 8, 9, 10, 11,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.06643v2-abstract-full').style.display = 'inline'; document.getElementById('2001.06643v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2001.06643v2-abstract-full" style="display: none;"> We extend a result of Guan by showing that the second Betti number of a 4-dimensional primitively symplectic orbifold is at most 23 and there are at most 91 singular points. The maximal possibility 23 can only occur in the smooth case. In addition to the known smooth examples with second Betti numbers 7 and 23, we provide examples of such orbifolds with second Betti numbers 3, 5, 6, 8, 9, 10, 11, 14 and 16. In an appendix, we extend Salamon's relation among Betti/Hodge numbers of symplectic manifolds to symplectic orbifolds. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.06643v2-abstract-full').style.display = 'none'; document.getElementById('2001.06643v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version. Published online in Mathematische Zeitschrift</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57R18; 14J28; 53C26; 19L10; 14B05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1911.11604">arXiv:1911.11604</a> <span> [<a href="https://arxiv.org/pdf/1911.11604">pdf</a>, <a href="https://arxiv.org/ps/1911.11604">ps</a>, <a href="https://arxiv.org/format/1911.11604">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Unirational Differential Curves and Differential Rational Parametrizations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lei Fu</a>, <a href="/search/math?searchtype=author&query=Li%2C+W">Wei Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1911.11604v2-abstract-short" style="display: inline;"> In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential curves. Then we show that the implicitization problem of proper linear differential rational parametric equations can be solved by means of differential resultants. F… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.11604v2-abstract-full').style.display = 'inline'; document.getElementById('1911.11604v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1911.11604v2-abstract-full" style="display: none;"> In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential curves. Then we show that the implicitization problem of proper linear differential rational parametric equations can be solved by means of differential resultants. Furthermore, for linear differential curves, we give an algorithm to determine whether an implicitly given linear differential curve is unirational and, in the affirmative case, to compute a proper differential rational parametrization for the differential curve. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.11604v2-abstract-full').style.display = 'none'; document.getElementById('1911.11604v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 November, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 12H05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1911.08399">arXiv:1911.08399</a> <span> [<a href="https://arxiv.org/pdf/1911.08399">pdf</a>, <a href="https://arxiv.org/ps/1911.08399">ps</a>, <a href="https://arxiv.org/format/1911.08399">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computational Physics">physics.comp-ph</span> </div> </div> <p class="title is-5 mathjax"> Time-Accurate and highly-Stable Explicit operators for stiff differential equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bassenne%2C+M">Maxime Bassenne</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Lin Fu</a>, <a href="/search/math?searchtype=author&query=Mani%2C+A">Ali Mani</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1911.08399v2-abstract-short" style="display: inline;"> Unconditionally stable implicit time-marching methods are powerful in solving stiff differential equations efficiently. In this work, a novel framework to handle stiff physical terms implicitly is proposed. Both physical and numerical stiffness originating from convection, diffusion and source terms (typically related to reaction) can be handled by a set of predefined Time-Accurate and highly-Stab… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.08399v2-abstract-full').style.display = 'inline'; document.getElementById('1911.08399v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1911.08399v2-abstract-full" style="display: none;"> Unconditionally stable implicit time-marching methods are powerful in solving stiff differential equations efficiently. In this work, a novel framework to handle stiff physical terms implicitly is proposed. Both physical and numerical stiffness originating from convection, diffusion and source terms (typically related to reaction) can be handled by a set of predefined Time-Accurate and highly-Stable Explicit (TASE) operators in a unified framework. The proposed TASE operators act as preconditioners on the stiff terms and can be deployed to any existing explicit time-marching methods straightforwardly. The resulting time integration methods remain the original explicit time-marching schemes, yet with nearly unconditional stability. The TASE operators can be designed to be arbitrarily high-order accurate with Richardson extrapolation such that the accuracy order of original explicit time-marching method is preserved. Theoretical analyses and stability diagrams show that the $s$-stages $s$th-order explicit Runge-Kutta (RK) methods are unconditionally stable when preconditioned by the TASE operators with order $p \leq s$ and $p \leq 2$. On the other hand, the $s$th-order RK methods preconditioned by the TASE operators with order of $p \leq s$ and $p > 2$ are nearly unconditionally stable. The only free parameter in TASE operators can be determined a priori based on stability arguments. Unlike classical implicit methods, the TASE methodology allows for solving non-linear problems with arbitrary order without requiring solving a nonlinear system of equations. A set of benchmark problems with strong stiffness is simulated to assess the performance of the TASE method. Numerical results suggest that the proposed framework preserves the high-order accuracy of the explicit time-marching methods with very-large time steps for all the considered cases. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.08399v2-abstract-full').style.display = 'none'; document.getElementById('1911.08399v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 November, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1911.06580">arXiv:1911.06580</a> <span> [<a href="https://arxiv.org/pdf/1911.06580">pdf</a>, <a href="https://arxiv.org/ps/1911.06580">ps</a>, <a href="https://arxiv.org/format/1911.06580">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s10231-021-01070-0">10.1007/s10231-021-01070-0 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Multiplicative Chow-K眉nneth decompositions and varieties of cohomological K3 type </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Laterveer%2C+R">Robert Laterveer</a>, <a href="/search/math?searchtype=author&query=Vial%2C+C">Charles Vial</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1911.06580v4-abstract-short" style="display: inline;"> Given a smooth projective variety, a Chow-K眉nneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-K盲hler varieties admit a multiplicative Chow-K眉nneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-K盲hler var… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.06580v4-abstract-full').style.display = 'inline'; document.getElementById('1911.06580v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1911.06580v4-abstract-full" style="display: none;"> Given a smooth projective variety, a Chow-K眉nneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-K盲hler varieties admit a multiplicative Chow-K眉nneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-K盲hler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow-K眉nneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and K眉chle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow-K眉nneth decomposition, by establishing this for two families of Todorov surfaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.06580v4-abstract-full').style.display = 'none'; document.getElementById('1911.06580v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 November, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages; updated numbering of sections to match published version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14C15; 14C25; 14C30; 14J45; 14J29 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ann. Mat. Pura Appl. (2021) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1911.06572">arXiv:1911.06572</a> <span> [<a href="https://arxiv.org/pdf/1911.06572">pdf</a>, <a href="https://arxiv.org/ps/1911.06572">ps</a>, <a href="https://arxiv.org/format/1911.06572">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0219199720500340">10.1142/S0219199720500340 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the motive of O'Grady's ten-dimensional hyper-K盲hler varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Floccari%2C+S">Salvatore Floccari</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Z">Ziyu Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1911.06572v4-abstract-short" style="display: inline;"> We investigate how the motive of hyper-K盲hler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.06572v4-abstract-full').style.display = 'inline'; document.getElementById('1911.06572v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1911.06572v4-abstract-full" style="display: none;"> We investigate how the motive of hyper-K盲hler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of B眉lles to the O'Grady-10 situation. In the non-commutative setting, similar results are proved for the Chow motive of moduli spaces of stable objects of the K3 category of a cubic fourfold. As a consequence, we provide abundant examples of hyper-K盲hler varieties of O'Grady-10 deformation type satisfying the standard conjectures. In the second part, we study the Andr茅 motive of projective hyper-K盲hler varieties. We attach to any such variety its defect group, an algebraic group which acts on the cohomology and measures the difference between the full motive and its weight-2 part. When the second Betti number is not 3, we show that the defect group is a natural complement of the Mumford--Tate group inside the motivic Galois group, and that it is deformation invariant. We prove the triviality of this group for all known examples of projective hyper-K盲hler varieties, so that in each case the full motive is controlled by its weight-2 part. As applications, we show that for any variety motivated by a product of known hyper-K盲hler varieties, all Hodge and Tate classes are motivated, the motivated Mumford--Tate conjecture holds, and the Andr茅 motive is abelian. This last point completes a recent work of Soldatenkov and provides a different proof for some of his results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.06572v4-abstract-full').style.display = 'none'; document.getElementById('1911.06572v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 July, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 November, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Published version. Communications in Contemporary Mathematics (open access)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14D20; 14C15; 14J28; 14F05; 14J32; 53C26 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Communications in Contemporary Mathematics, 2020 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1910.08107">arXiv:1910.08107</a> <span> [<a href="https://arxiv.org/pdf/1910.08107">pdf</a>, <a href="https://arxiv.org/format/1910.08107">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Heterocedasticity-Adjusted Ranking and Thresholding for Large-Scale Multiple Testing </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Luella Fu</a>, <a href="/search/math?searchtype=author&query=Gang%2C+B">Bowen Gang</a>, <a href="/search/math?searchtype=author&query=James%2C+G+M">Gareth M. James</a>, <a href="/search/math?searchtype=author&query=Sun%2C+W">Wenguang Sun</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1910.08107v2-abstract-short" style="display: inline;"> Standardization has been a widely adopted practice in multiple testing, for it takes into account the variability in sampling and makes the test statistics comparable across different study units. However, despite conventional wisdom to the contrary, we show that there can be a significant loss in information from basing hypothesis tests on standardized statistics rather than the full data. We dev… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.08107v2-abstract-full').style.display = 'inline'; document.getElementById('1910.08107v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1910.08107v2-abstract-full" style="display: none;"> Standardization has been a widely adopted practice in multiple testing, for it takes into account the variability in sampling and makes the test statistics comparable across different study units. However, despite conventional wisdom to the contrary, we show that there can be a significant loss in information from basing hypothesis tests on standardized statistics rather than the full data. We develop a new class of heteroscedasticity--adjusted ranking and thresholding (HART) rules that aim to improve existing methods by simultaneously exploiting commonalities and adjusting heterogeneities among the study units. The main idea of HART is to bypass standardization by directly incorporating both the summary statistic and its variance into the testing procedure. A key message is that the variance structure of the alternative distribution, which is subsumed under standardized statistics, is highly informative and can be exploited to achieve higher power. The proposed HART procedure is shown to be asymptotically valid and optimal for false discovery rate (FDR) control. Our simulation results demonstrate that HART achieves substantial power gain over existing methods at the same FDR level. We illustrate the implementation through a microarray analysis of myeloma. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.08107v2-abstract-full').style.display = 'none'; document.getElementById('1910.08107v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 March, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 October, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">55 pages, 13 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1907.10868">arXiv:1907.10868</a> <span> [<a href="https://arxiv.org/pdf/1907.10868">pdf</a>, <a href="https://arxiv.org/ps/1907.10868">ps</a>, <a href="https://arxiv.org/format/1907.10868">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> A motivic global Torelli theorem for isogenous K3 surfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Vial%2C+C">Charles Vial</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1907.10868v2-abstract-short" style="display: inline;"> We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective K3 surfaces are isogenous (i.e. their second rational cohomology groups are Hodge isometric) if and only if their Chow motives are isomorphic… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1907.10868v2-abstract-full').style.display = 'inline'; document.getElementById('1907.10868v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1907.10868v2-abstract-full" style="display: none;"> We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective K3 surfaces are isogenous (i.e. their second rational cohomology groups are Hodge isometric) if and only if their Chow motives are isomorphic as Frobenius algebra objects; this can be regarded as a motivic Torelli-type theorem. We ask whether, more generally, twisted derived equivalent hyper-Kaehler varieties have isomorphic Chow motives as (Frobenius) algebra objects and in particular isomorphic graded rational cohomology algebras. In the appendix, we justify introducing the notion of "Frobenius algebra object" by showing the existence of an infinite family of K3 surfaces whose Chow motives are pairwise non-isomorphic as Frobenius algebra objects but isomorphic as algebra objects. In particular, K3 surfaces in that family are pairwise non-isogenous but have isomorphic rational Hodge algebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1907.10868v2-abstract-full').style.display = 'none'; document.getElementById('1907.10868v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 July, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">To appear in Advances in Math. The published version will not contain Section A.5</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14C25; 14C34; 14C15; 14F05; 14J28; 14J32 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1810.11873">arXiv:1810.11873</a> <span> [<a href="https://arxiv.org/pdf/1810.11873">pdf</a>, <a href="https://arxiv.org/ps/1810.11873">ps</a>, <a href="https://arxiv.org/format/1810.11873">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00029-019-0482-y">10.1007/s00029-019-0482-y <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Hilbert squares: derived categories and deformations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Belmans%2C+P">Pieter Belmans</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Raedschelders%2C+T">Theo Raedschelders</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1810.11873v2-abstract-short" style="display: inline;"> For a smooth projective variety $X$ with exceptional structure sheaf, and $\operatorname{Hilb}^2X$ the Hilbert scheme of two points on $X$, we show that the Fourier-Mukai functor $\mathbf{D}^{\mathrm{b}}(X) \to\mathbf{D}^{\mathrm{b}}(\operatorname{Hilb}^2X)$ induced by the universal ideal sheaf is fully faithful, provided the dimension of $X$ is at least 2. This fully faithfulness allows us to con… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.11873v2-abstract-full').style.display = 'inline'; document.getElementById('1810.11873v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1810.11873v2-abstract-full" style="display: none;"> For a smooth projective variety $X$ with exceptional structure sheaf, and $\operatorname{Hilb}^2X$ the Hilbert scheme of two points on $X$, we show that the Fourier-Mukai functor $\mathbf{D}^{\mathrm{b}}(X) \to\mathbf{D}^{\mathrm{b}}(\operatorname{Hilb}^2X)$ induced by the universal ideal sheaf is fully faithful, provided the dimension of $X$ is at least 2. This fully faithfulness allows us to construct a spectral sequence relating the deformation theories of $X$ and $\operatorname{Hilb}^2X$ and to show that it degenerates at the second page, giving a Hochschild-Kostant-Rosenberg-type filtration on the Hochschild cohomology of $X$. These results generalise known results for surfaces due to Krug-Sosna, Fantechi and Hitchin. Finally, as a by-product, we discover the following surprising phenomenon: for a smooth projective variety of dimension at least 3 with exceptional structure sheaf, it is rigid if and only if its Hilbert scheme of two points is rigid. This last fact contrasts drastically to the surface case: non-commutative deformations of a surface contribute to commutative deformations of its Hilbert square. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.11873v2-abstract-full').style.display = 'none'; document.getElementById('1810.11873v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages, all comments welcome, added a reference to a recent faithfulness result</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14F05; 14J10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Selecta Math. (N.S.) 25 (2019), no. 3, 37 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1809.03710">arXiv:1809.03710</a> <span> [<a href="https://arxiv.org/pdf/1809.03710">pdf</a>, <a href="https://arxiv.org/ps/1809.03710">ps</a>, <a href="https://arxiv.org/format/1809.03710">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="K-Theory and Homology">math.KT</span> </div> </div> <p class="title is-5 mathjax"> Orbifold products for higher K-theory and motivic cohomology </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Nguyen%2C+M+T">Manh Toan Nguyen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1809.03710v2-abstract-short" style="display: inline;"> Due to the work of many authors in the last decades, given an algebraic orbifold (smooth proper Deligne-Mumford stack with trivial generic stabilizer), one can construct its orbifold Chow ring and orbifold Grothendieck ring, and relate them by the orbifold Chern character map, generalizing the fundamental work of Chen-Ruan on orbifold cohomology. In this paper, we extend this theory naturally to h… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1809.03710v2-abstract-full').style.display = 'inline'; document.getElementById('1809.03710v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1809.03710v2-abstract-full" style="display: none;"> Due to the work of many authors in the last decades, given an algebraic orbifold (smooth proper Deligne-Mumford stack with trivial generic stabilizer), one can construct its orbifold Chow ring and orbifold Grothendieck ring, and relate them by the orbifold Chern character map, generalizing the fundamental work of Chen-Ruan on orbifold cohomology. In this paper, we extend this theory naturally to higher Chow groups and higher algebraic K-theory, mainly following the work of Jarvis-Kaufmann-Kimura and Edidin-Jarvis-Kimura. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1809.03710v2-abstract-full').style.display = 'none'; document.getElementById('1809.03710v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 October, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 September, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version to appear in Documenta Mathematica</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 19E08; 19E15; 14C15; 55N32 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1808.05851">arXiv:1808.05851</a> <span> [<a href="https://arxiv.org/pdf/1808.05851">pdf</a>, <a href="https://arxiv.org/ps/1808.05851">ps</a>, <a href="https://arxiv.org/format/1808.05851">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Supersingular irreducible symplectic varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Li%2C+Z">Zhiyuan Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1808.05851v5-abstract-short" style="display: inline;"> We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of positive characteristic, under natural numerical conditions: (1) smooth moduli spaces of sheaves on K3… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.05851v5-abstract-full').style.display = 'inline'; document.getElementById('1808.05851v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1808.05851v5-abstract-full" style="display: none;"> We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of positive characteristic, under natural numerical conditions: (1) smooth moduli spaces of sheaves on K3 surfaces and (2) smooth Albanese fibers of moduli spaces of sheaves on abelian surfaces. Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to be equivalent in both cases (1) and (2). Their equivalence is conjectured in general. On the geometric aspect, we conjecture that unirationality characterizes supersingularity for symplectic varieties. Such an equivalence is established in case (1), assuming the same is true for K3 surfaces. In case (2), we show that the rational chain connectedness is equivalent to supersingularity. On the motivic aspect, we conjecture that the algebraic cycles on supersingular symplectic varieties should be much simpler than their complex counterparts: its rational Chow motive is of supersingular abelian type, the rational Chow ring is representable and satisfies the Bloch--Beilinson conjecture and Beauville's splitting property. As evidences, we prove all these conjectures on algebraic cycles for supersingular varieties in both cases (1) and (2). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.05851v5-abstract-full').style.display = 'none'; document.getElementById('1808.05851v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 August, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version. To appear in "Rationality of Algebraic Varieties", proceedings of the 2019 Schiermonnikoog conference</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14J28; 14J42; 14J60; 14C15; 14C25; 14M20; 14C15; 14G17; 14K99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1806.03864">arXiv:1806.03864</a> <span> [<a href="https://arxiv.org/pdf/1806.03864">pdf</a>, <a href="https://arxiv.org/ps/1806.03864">ps</a>, <a href="https://arxiv.org/format/1806.03864">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00208-019-01876-7">10.1007/s00208-019-01876-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Finiteness of Klein actions and real structures on compact hyperk盲hler manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cattaneo%2C+A">Andrea Cattaneo</a>, <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1806.03864v3-abstract-short" style="display: inline;"> One central problem in real algebraic geometry is to classify the real structures of a given complex manifold. We address this problem for compact hyperk盲hler manifolds by showing that any such manifold admits only finitely many real structures up to equivalence. We actually prove more generally that there are only finitely many, up to conjugacy, faithful finite group actions by holomorphic or ant… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1806.03864v3-abstract-full').style.display = 'inline'; document.getElementById('1806.03864v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1806.03864v3-abstract-full" style="display: none;"> One central problem in real algebraic geometry is to classify the real structures of a given complex manifold. We address this problem for compact hyperk盲hler manifolds by showing that any such manifold admits only finitely many real structures up to equivalence. We actually prove more generally that there are only finitely many, up to conjugacy, faithful finite group actions by holomorphic or anti-holomorphic automorphisms (the so-called Klein actions). In other words, the automorphism group and the Klein automorphism group of a compact hyperk盲hler manifold contain only finitely many conjugacy classes of finite subgroups. We furthermore answer a question of Oguiso by showing that the automorphism group of a compact hyperk盲hler manifold is finitely presented. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1806.03864v3-abstract-full').style.display = 'none'; document.getElementById('1806.03864v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 August, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version, published in Mathematische Annalen</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14P99; 14J50; 53G26 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mathematische Annalen, 2019 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1804.05297">arXiv:1804.05297</a> <span> [<a href="https://arxiv.org/pdf/1804.05297">pdf</a>, <a href="https://arxiv.org/ps/1804.05297">ps</a>, <a href="https://arxiv.org/format/1804.05297">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> The $p$-adic Gelfand-Kapranov-Zelevinsky hypergeometric complex </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lei Fu</a>, <a href="/search/math?searchtype=author&query=Li%2C+P">Peigen Li</a>, <a href="/search/math?searchtype=author&query=Wan%2C+D">Daqing Wan</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Hao Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1804.05297v3-abstract-short" style="display: inline;"> To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $p$-adic counterpart of the GKZ hypergeometric system. The $p$-adic GKZ hypergeometric complex is a twisted relative de Rham complex of over-convergent differentia… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1804.05297v3-abstract-full').style.display = 'inline'; document.getElementById('1804.05297v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1804.05297v3-abstract-full" style="display: none;"> To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $p$-adic counterpart of the GKZ hypergeometric system. The $p$-adic GKZ hypergeometric complex is a twisted relative de Rham complex of over-convergent differential forms with logarithmic poles. It is an over-holonomic object in the derived category of arithmetic $\mathcal D$-modules with Frobenius structures. Traces of Frobenius on fibers at Techm眉ller points of the GKZ hypergeometric complex define the hypergeometric function over the finite field introduced by Gelfand and Graev. Over the non-degenerate locus, the GKZ hypergeometric complex defines an over-convergent $F$-isocrystal. It is the crystalline companion of the $\ell$-adic GKZ hypergeometric sheaf that we constructed before. Our method is a combination of Dwork's theory and the theory of arithmetic $\mathcal D$-modules of Berthelot. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1804.05297v3-abstract-full').style.display = 'none'; document.getElementById('1804.05297v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version. To appear in Mathematische Annalen</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14F30; 11T23; 14G15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1801.03995">arXiv:1801.03995</a> <span> [<a href="https://arxiv.org/pdf/1801.03995">pdf</a>, <a href="https://arxiv.org/ps/1801.03995">ps</a>, <a href="https://arxiv.org/format/1801.03995">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> 2-cycles sur les hypersurfaces cubiques de dimension 5 </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Tian%2C+Z">Zhiyu Tian</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1801.03995v2-abstract-short" style="display: inline;"> On 茅tudie les cycles alg茅briques de codimension 3 sur les hypersurfaces cubiques lisses de dimension 5. Pour une telle hypersurface, on d茅montre d'une part que son groupe de Griffiths des cycles de codimension 3 est trivial et d'autre part que l'application d'Abel-Jacobi induit un isomorphisme entre son groupe de Chow des cycles de codimension 3 alg茅briquement equivalents 脿 z茅ro et sa jacobienne i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.03995v2-abstract-full').style.display = 'inline'; document.getElementById('1801.03995v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1801.03995v2-abstract-full" style="display: none;"> On 茅tudie les cycles alg茅briques de codimension 3 sur les hypersurfaces cubiques lisses de dimension 5. Pour une telle hypersurface, on d茅montre d'une part que son groupe de Griffiths des cycles de codimension 3 est trivial et d'autre part que l'application d'Abel-Jacobi induit un isomorphisme entre son groupe de Chow des cycles de codimension 3 alg茅briquement equivalents 脿 z茅ro et sa jacobienne interm茅diaire. ---------- We study 2-cycles of a smooth cubic hypersurface of dimension 5. We show that the Griffiths group of 2-cycles is trivial and the Abel-Jacobi map induces an isomorphism between the Chow group of algebraically trivial 2-cycles and the intermediate Jacobian. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.03995v2-abstract-full').style.display = 'none'; document.getElementById('1801.03995v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 January, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">In French. Final version. To appear in Mathematische Zeitschrift</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1709.05644">arXiv:1709.05644</a> <span> [<a href="https://arxiv.org/pdf/1709.05644">pdf</a>, <a href="https://arxiv.org/format/1709.05644">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Distinguished cycles on varieties with motive of abelian type and the Section Property </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Vial%2C+C">Charles Vial</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1709.05644v4-abstract-short" style="display: inline;"> A remarkable result of Peter O'Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville's splitting principle, we formulate a conjectural Section Property which predicts that for smooth projective holomorphic symplectic varieties there exists such a sec… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.05644v4-abstract-full').style.display = 'inline'; document.getElementById('1709.05644v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.05644v4-abstract-full" style="display: none;"> A remarkable result of Peter O'Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville's splitting principle, we formulate a conjectural Section Property which predicts that for smooth projective holomorphic symplectic varieties there exists such a section of algebra whose image contains all the Chern classes of the variety. In this paper, we investigate this property for (not necessarily symplectic) varieties with Chow motive of abelian type. We introduce the notion of symmetrically distinguished abelian motive and use it to provide a sufficient condition for a smooth projective variety to admit such a section. We then give series of examples of varieties for which our theory works. For instance, we prove the existence of such a section for arbitrary products of varieties with Chow groups of finite rank, abelian varieties, hyperelliptic curves, Fermat cubic hypersurfaces, Hilbert schemes of points on an abelian surface or a Kummer surface or a K3 surface with Picard number at least 19, and generalized Kummer varieties. The latter cases provide evidence for the conjectural Section Property and exemplify the mantra that the motives of holomorphic symplectic varieties should behave as the motives of abelian varieties, as algebra objects. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.05644v4-abstract-full').style.display = 'none'; document.getElementById('1709.05644v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 July, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">44 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14C05; 14C25; 14C15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1709.01714">arXiv:1709.01714</a> <span> [<a href="https://arxiv.org/pdf/1709.01714">pdf</a>, <a href="https://arxiv.org/ps/1709.01714">ps</a>, <a href="https://arxiv.org/format/1709.01714">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Motivic multiplicative McKay correspondence for surfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Tian%2C+Z">Zhiyu Tian</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1709.01714v3-abstract-short" style="display: inline;"> We revisit the classical two-dimensional McKay correspondence in two respects: The first one, which is the main point of this work, is that we take into account of the multiplicative structure given by the orbifold product; second, instead of using cohomology, we deal with the Chow motives. More precisely, we prove that for any smooth proper two-dimensional orbifold with projective coarse moduli s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.01714v3-abstract-full').style.display = 'inline'; document.getElementById('1709.01714v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.01714v3-abstract-full" style="display: none;"> We revisit the classical two-dimensional McKay correspondence in two respects: The first one, which is the main point of this work, is that we take into account of the multiplicative structure given by the orbifold product; second, instead of using cohomology, we deal with the Chow motives. More precisely, we prove that for any smooth proper two-dimensional orbifold with projective coarse moduli space, there is an isomorphism of algebra objects, in the category of complex Chow motives, between the motive of the minimal resolution and the orbifold motive. In particular, the complex Chow ring (resp. Grothendieck ring, cohomology ring, topological K-theory) of the minimal resolution is isomorphic to the complex orbifold Chow ring (resp. Grothendieck ring, cohomology ring, topological K-theory) of the orbifold surface. This confirms the two-dimensional Motivic Crepant Resolution Conjecture. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.01714v3-abstract-full').style.display = 'none'; document.getElementById('1709.01714v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages. Final version. To appear in Manuscripta Mathematica</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14E16; 14C15; 14J17; 55N32 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1708.02919">arXiv:1708.02919</a> <span> [<a href="https://arxiv.org/pdf/1708.02919">pdf</a>, <a href="https://arxiv.org/ps/1708.02919">ps</a>, <a href="https://arxiv.org/format/1708.02919">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> The generalized Franchetta conjecture for some hyper-K盲hler varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lie Fu</a>, <a href="/search/math?searchtype=author&query=Laterveer%2C+R">Robert Laterveer</a>, <a href="/search/math?searchtype=author&query=Vial%2C+C">Charles Vial</a>, <a href="/search/math?searchtype=author&query=Shen%2C+M">Mingmin Shen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1708.02919v3-abstract-short" style="display: inline;"> The generalized Franchetta conjecture for hyper-K盲hler varieties predicts that an algebraic cycle on the universal family of certain polarized hyper-K盲hler varieties is fiberwise rationally equivalent to zero if and only if it vanishes in cohomology fiberwise. We establish Franchetta-type results for certain low (Hilbert) powers of low degree K3 surfaces, for the Beauville--Donagi family of Fano v… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1708.02919v3-abstract-full').style.display = 'inline'; document.getElementById('1708.02919v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1708.02919v3-abstract-full" style="display: none;"> The generalized Franchetta conjecture for hyper-K盲hler varieties predicts that an algebraic cycle on the universal family of certain polarized hyper-K盲hler varieties is fiberwise rationally equivalent to zero if and only if it vanishes in cohomology fiberwise. We establish Franchetta-type results for certain low (Hilbert) powers of low degree K3 surfaces, for the Beauville--Donagi family of Fano varieties of lines on cubic fourfolds and its relative square, and for $0$-cycles and codimension-$2$ cycles for the Lehn--Lehn--Sorger--van Straten family of hyper-K盲hler eightfolds. We also draw many consequences in the direction of the Beauville--Voisin conjecture as well as Voisin's refinement involving coisotropic subvarieties. In the appendix, we establish a new relation among tautological cycles on the square of the Fano variety of lines of a smooth cubic fourfold and provide some applications. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1708.02919v3-abstract-full').style.display = 'none'; document.getElementById('1708.02919v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 August, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">35 pages; to appear in J. Math. Pures Appl</span> </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Fu%2C+L&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Fu%2C+L&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Fu%2C+L&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> </ul> </nav> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul 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