Endoscopic decompositions and the Hausel–Thaddeus conjecture
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endoscopic-decompositions-and-the-hauselthaddeus-conjecture.pdf
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Published version
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713.11 KB
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Author(s)
Maulik, Davesh
Shen, Junliang
Date Issued
2021
Journal
Forum of Mathematics, Pi
Publisher
Cambridge University Press (CUP)
Citation
Maulik, Davesh and Shen, Junliang. 2021. "Endoscopic decompositions and the Hausel–Thaddeus conjecture." Forum of Mathematics, Pi, 9.
Version
Final published version
Abstract
Abstract
We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover the topological mirror symmetry conjecture of Hausel and Thaddeus concerning $\mathrm {SL}_n$ - and $\mathrm {PGL}_n$ -Higgs bundles. This provides a complete description of the cohomology of the moduli space of stable $\mathrm {SL}_n$ -Higgs bundles in terms of the tautological classes, and gives a new proof of the Hausel–Thaddeus conjecture, which was also proven recently by Gröchenig, Wyss and Ziegler via p-adic integration. Our method is to relate the decomposition theorem for the Hitchin fibration, using vanishing cycle functors, to the decomposition theorem for the twisted Hitchin fibration, whose supports are simpler.
We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover the topological mirror symmetry conjecture of Hausel and Thaddeus concerning $\mathrm {SL}_n$ - and $\mathrm {PGL}_n$ -Higgs bundles. This provides a complete description of the cohomology of the moduli space of stable $\mathrm {SL}_n$ -Higgs bundles in terms of the tautological classes, and gives a new proof of the Hausel–Thaddeus conjecture, which was also proven recently by Gröchenig, Wyss and Ziegler via p-adic integration. Our method is to relate the decomposition theorem for the Hitchin fibration, using vanishing cycle functors, to the decomposition theorem for the twisted Hitchin fibration, whose supports are simpler.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution 4.0 International license
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DOI of Published Version
https://doi.org/10.1017/fmp.2021.7
