[2011.04603] On character varieties of singular manifolds

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This construction is valid for any algebraic group $G$, in any dimension and also in the parabolic setting. In particular, this TQFT allow us to compute the virtual classes of representation varieties over complex singular planar curves. In addition, in the case $G = \mathrm{SL}_{2}(k)$, the virtual class of the associated character variety over a nodal closed orientable surface is computed both in the non-parabolic and in the parabolic scenarios."/> <meta name="twitter:site" content="@arxiv"/> <meta name="twitter:card" content="summary"/> <meta name="twitter:title" content="On character varieties of singular manifolds"/> <meta name="twitter:description" content="In this paper, we construct a lax monoidal Topological Quantum Field Theory that computes virtual classes, in the Grothendieck ring of algebraic varieties, of $G$-representation varieties over..."/> <meta name="twitter:image" content="https://static.arxiv.org/icons/twitter/arxiv-logo-twitter-square.png"/> <meta name="twitter:image:alt" content="arXiv logo"/> <link rel="stylesheet" media="screen" type="text/css" href="/static/browse/0.3.4/css/tooltip.css"/><link rel="stylesheet" media="screen" type="text/css" href="https://static.arxiv.org/js/bibex-dev/bibex.css?20200709"/> <script src="/static/browse/0.3.4/js/mathjaxToggle.min.js" type="text/javascript"></script> <script src="//code.jquery.com/jquery-latest.min.js" type="text/javascript"></script> <script src="//cdn.jsdelivr.net/npm/js-cookie@2/src/js.cookie.min.js" type="text/javascript"></script> <script src="//cdn.jsdelivr.net/npm/dompurify@2.3.5/dist/purify.min.js"></script> <script src="/static/browse/0.3.4/js/toggle-labs.js?20241022" type="text/javascript"></script> <script src="/static/browse/0.3.4/js/cite.js" type="text/javascript"></script><meta name="citation_title" content="On character varieties of singular manifolds" /><meta name="citation_author" content="Gonz谩lez-Prieto, 脕ngel" /><meta name="citation_author" content="Logares, Marina" /><meta name="citation_date" content="2020/11/09" /><meta name="citation_online_date" content="2020/11/09" /><meta name="citation_pdf_url" content="http://arxiv.org/pdf/2011.04603" /><meta name="citation_arxiv_id" content="2011.04603" /><meta name="citation_abstract" content="In this paper, we construct a lax monoidal Topological Quantum Field Theory that computes virtual classes, in the Grothendieck ring of algebraic varieties, of $G$-representation varieties over manifolds with conic singularities, which we will call nodefolds. This construction is valid for any algebraic group $G$, in any dimension and also in the parabolic setting. In particular, this TQFT allow us to compute the virtual classes of representation varieties over complex singular planar curves. In addition, in the case $G = \mathrm{SL}_{2}(k)$, the virtual class of the associated character variety over a nodal closed orientable surface is computed both in the non-parabolic and in the parabolic scenarios." /> </head> <body class="with-cu-identity"> <div class="flex-wrap-footer"> <header> <a href="#content" class="is-sr-only">Skip to main content</a>  <div class="columns is-vcentered is-hidden-mobile" id="cu-identity"> <div class="column" id="cu-logo"> <a href="https://www.cornell.edu/"><img src="/static/browse/0.3.4/images/icons/cu/cornell-reduced-white-SMALL.svg" alt="Cornell University" /></a> </div><div class="column" id="support-ack"> <span id="support-ack-url">We gratefully acknowledge support from the Simons Foundation, <a href="https://info.arxiv.org/about/ourmembers.html">member institutions</a>, and all contributors.</span> <a href="https://info.arxiv.org/about/donate.html" class="btn-header-donate">Donate</a> </div> </div> <div id="header" class="is-hidden-mobile"> 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href="https://arxiv.org/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto,+%C3%81" rel="nofollow">脕ngel Gonz谩lez-Prieto</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Logares,+M" rel="nofollow">Marina Logares</a></div> <div id="download-button-info" hidden>View a PDF of the paper titled On character varieties of singular manifolds, by \'Angel Gonz\'alez-Prieto and 1 other authors</div> <a class="mobile-submission-download" href="/pdf/2011.04603">View PDF</a> <blockquote class="abstract mathjax"> <span class="descriptor">Abstract:</span>In this paper, we construct a lax monoidal Topological Quantum Field Theory that computes virtual classes, in the Grothendieck ring of algebraic varieties, of $G$-representation varieties over manifolds with conic singularities, which we will call nodefolds. This construction is valid for any algebraic group $G$, in any dimension and also in the parabolic setting. In particular, this TQFT allow us to compute the virtual classes of representation varieties over complex singular planar curves. In addition, in the case $G = \mathrm{SL}_{2}(k)$, the virtual class of the associated character variety over a nodal closed orientable surface is computed both in the non-parabolic and in the parabolic scenarios. </blockquote>  <div class="metatable"> <table summary="Additional metadata"> <tr> <td class="tablecell label">Comments:</td> <td class="tablecell comments mathjax">30 pages, 4 figures</td> </tr> <tr> <td class="tablecell label">Subjects:</td> <td class="tablecell subjects"> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Category Theory (math.CT); Representation Theory (math.RT)</td> </tr> <tr> <td class="tablecell label"><abbr title="Mathematical Subject Classification">MSC</abbr> classes:</td> <td class="tablecell msc-classes">57R56 (Primary), 19E08, 32S50, 14D21, 20G05 (Secondary)</td> </tr> <tr> <td class="tablecell label">Cite as:</td> <td class="tablecell arxivid"><span class="arxivid"><a href="https://arxiv.org/abs/2011.04603">arXiv:2011.04603</a> [math.AG]</span></td> </tr> <tr> <td class="tablecell label"> </td> <td class="tablecell arxividv">(or <span class="arxivid"> <a href="https://arxiv.org/abs/2011.04603v1">arXiv:2011.04603v1</a> [math.AG]</span> for this version) </td> </tr> <tr> <td class="tablecell label"> </td> <td class="tablecell arxivdoi"> <a href="https://doi.org/10.48550/arXiv.2011.04603" id="arxiv-doi-link">https://doi.org/10.48550/arXiv.2011.04603</a><div class="button-and-tooltip"> <button class="more-info" aria-describedby="more-info-desc-1"> <svg height="15" role="presentation" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256 8C119.043 8 8 119.083 8 256c0 136.997 111.043 248 248 248s248-111.003 248-248C504 119.083 392.957 8 256 8zm0 110c23.196 0 42 18.804 42 42s-18.804 42-42 42-42-18.804-42-42 18.804-42 42-42zm56 254c0 6.627-5.373 12-12 12h-88c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h12v-64h-12c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h64c6.627 0 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[2011.04603] On character varieties of singular manifolds