Hitchin fibrations, abelian surfaces, and the P=W conjecture
<p>We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics>...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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American Mathematical Society (AMS)
2022
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| Online Access: | https://hdl.handle.net/1721.1/145788 |
| _version_ | 1826189765535334400 |
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| author | de Cataldo, Mark Maulik, Davesh Shen, Junliang |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics de Cataldo, Mark Maulik, Davesh Shen, Junliang |
| author_sort | de Cataldo, Mark |
| collection | MIT |
| description | <p>We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2">
<mml:semantics>
<mml:mn>2</mml:mn>
<mml:annotation encoding="application/x-tex">2</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration.</p>
<p>Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert–Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markman’s monodromy operators play a crucial role.</p> |
| first_indexed | 2024-09-23T08:21:12Z |
| format | Article |
| id | mit-1721.1/145788 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T08:21:12Z |
| publishDate | 2022 |
| publisher | American Mathematical Society (AMS) |
| record_format | dspace |
| spelling | mit-1721.1/1457882022-10-13T03:46:06Z Hitchin fibrations, abelian surfaces, and the P=W conjecture de Cataldo, Mark Maulik, Davesh Shen, Junliang Massachusetts Institute of Technology. Department of Mathematics <p>We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration.</p> <p>Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert–Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markman’s monodromy operators play a crucial role.</p> 2022-10-12T14:06:02Z 2022-10-12T14:06:02Z 2021-11-02 2022-10-12T13:57:02Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/145788 de Cataldo, Mark, Maulik, Davesh and Shen, Junliang. 2021. "Hitchin fibrations, abelian surfaces, and the P=W conjecture." Journal of the American Mathematical Society, 35 (3). en 10.1090/jams/989 Journal of the American Mathematical Society Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf American Mathematical Society (AMS) American Mathematical Society |
| spellingShingle | de Cataldo, Mark Maulik, Davesh Shen, Junliang Hitchin fibrations, abelian surfaces, and the P=W conjecture |
| title | Hitchin fibrations, abelian surfaces, and the P=W conjecture |
| title_full | Hitchin fibrations, abelian surfaces, and the P=W conjecture |
| title_fullStr | Hitchin fibrations, abelian surfaces, and the P=W conjecture |
| title_full_unstemmed | Hitchin fibrations, abelian surfaces, and the P=W conjecture |
| title_short | Hitchin fibrations, abelian surfaces, and the P=W conjecture |
| title_sort | hitchin fibrations abelian surfaces and the p w conjecture |
| url | https://hdl.handle.net/1721.1/145788 |
| work_keys_str_mv | AT decataldomark hitchinfibrationsabeliansurfacesandthepwconjecture AT maulikdavesh hitchinfibrationsabeliansurfacesandthepwconjecture AT shenjunliang hitchinfibrationsabeliansurfacesandthepwconjecture |