[1807.08540] Pseudo-quotients of algebraic actions and their application to character varieties

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They arise when we focus on the purely topological properties of good GIT quotients regardless of their algebraic properties. The flexibility granted by their topological nature enables an easier identification in geometric constructions than classical GIT quotients. We obtain several results about the interplay between pseudo-quotients and good quotients. Additionally, we show that in characteristic zero pseudo-quotients are unique up to virtual class in the Grothendieck ring of algebraic varieties. As an application, we compute the virtual class of $\mathrm{SL}_{2}(k)$-character varieties for free groups and surface groups as well as their parabolic counterparts with punctures of Jordan type."/> <meta name="twitter:site" content="@arxiv"/> <meta name="twitter:card" content="summary"/> <meta name="twitter:title" content="Pseudo-quotients of algebraic actions and their application to..."/> <meta name="twitter:description" content="In this paper, we propose a weak version of quotient for the algebraic action of a group on a variety, which we shall call a pseudo-quotient. 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They arise when we focus on the purely topological properties of good GIT quotients regardless of their algebraic properties. The flexibility granted by their topological nature enables an easier identification in geometric constructions than classical GIT quotients. We obtain several results about the interplay between pseudo-quotients and good quotients. Additionally, we show that in characteristic zero pseudo-quotients are unique up to virtual class in the Grothendieck ring of algebraic varieties. As an application, we compute the virtual class of $\mathrm{SL}_{2}(k)$-character varieties for free groups and surface groups as well as their parabolic counterparts with punctures of Jordan type." /> </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"> <a 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varieties</h1> <div class="authors"><span class="descriptor">Authors:</span><a href="https://arxiv.org/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto,+%C3%81" rel="nofollow">脕ngel Gonz谩lez-Prieto</a></div> <div id="download-button-info" hidden>View a PDF of the paper titled Pseudo-quotients of algebraic actions and their application to character varieties, by \'Angel Gonz\'alez-Prieto</div> <a class="mobile-submission-download" href="/pdf/1807.08540">View PDF</a> <blockquote class="abstract mathjax"> <span class="descriptor">Abstract:</span>In this paper, we propose a weak version of quotient for the algebraic action of a group on a variety, which we shall call a pseudo-quotient. They arise when we focus on the purely topological properties of good GIT quotients regardless of their algebraic properties. The flexibility granted by their topological nature enables an easier identification in geometric constructions than classical GIT quotients. We obtain several results about the interplay between pseudo-quotients and good quotients. Additionally, we show that in characteristic zero pseudo-quotients are unique up to virtual class in the Grothendieck ring of algebraic varieties. As an application, we compute the virtual class of $\mathrm{SL}_{2}(k)$-character varieties for free groups and surface groups as well as their parabolic counterparts with punctures of Jordan type. </blockquote>  <div class="metatable"> <table summary="Additional metadata"> <tr> <td class="tablecell label">Comments:</td> <td class="tablecell comments mathjax">39 pages, no figures. Final version</td> </tr> <tr> <td class="tablecell label">Subjects:</td> <td class="tablecell subjects"> <span class="primary-subject">Algebraic Geometry (math.AG)</span>; Geometric Topology (math.GT); 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">14L24 (Primary) 14C25, 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/1807.08540">arXiv:1807.08540</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/1807.08540v5">arXiv:1807.08540v5</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.1807.08540" id="arxiv-doi-link">https://doi.org/10.48550/arXiv.1807.08540</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 12 5.373 12 12v100h12c6.627 0 12 5.373 12 12v24z" class=""></path></svg> <span class="visually-hidden">Focus to learn more</span> </button>  <div role="tooltip" id="more-info-desc-1"> <span class="left-corner"></span> arXiv-issued DOI via DataCite</div> </div> </td> </tr> <tr> <td class="tablecell label">Journal reference:</td> <td class="tablecell jref">Communications in Contemporary Mathematics (2023)</td> </tr> <tr> <td class="tablecell label"> <abbr title="Digital Object Identifier">Related DOI</abbr>: </td> <td class="tablecell doi"><a href="https://doi.org/10.1142/S0219199723500098" data-doi="10.1142/S0219199723500098" class="link-https link-external" rel="external noopener nofollow">https://doi.org/10.1142/S0219199723500098</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 12 5.373 12 12v100h12c6.627 0 12 5.373 12 12v24z" class=""></path></svg> <span class="visually-hidden">Focus to learn more</span> </button>  <div role="tooltip" id="more-info-desc-1"> <span class="left-corner"></span> DOI(s) linking to related resources </div> </div> </td> </tr> </table> </div> </div> </div> <div class="submission-history"> <h2>Submission history</h2> From: 脕ngel Gonz谩lez-Prieto Dr. [<a href="/show-email/53542f36/1807.08540" rel="nofollow">view email</a>] <br/> <strong><a href="/abs/1807.08540v1" rel="nofollow">[v1]</a></strong> Mon, 23 Jul 2018 11:36:52 UTC (43 KB)<br/> <strong><a href="/abs/1807.08540v2" rel="nofollow">[v2]</a></strong> Tue, 13 Nov 2018 17:37:33 UTC (34 KB)<br/> <strong><a href="/abs/1807.08540v3" rel="nofollow">[v3]</a></strong> Wed, 2 Sep 2020 21:16:24 UTC (41 KB)<br/> <strong><a href="/abs/1807.08540v4" rel="nofollow">[v4]</a></strong> Fri, 16 Apr 2021 13:38:26 UTC (51 KB)<br/> <strong>[v5]</strong> Thu, 2 Nov 2023 00:26:43 UTC (46 KB)<br/> </div> </div>  <div 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[1807.08540] Pseudo-quotients of algebraic actions and their application to character varieties