Motivic integration on the Hitchin fibration

Loeser, François; Wyss, Dimitri

doi:10.14231/ag-2021-004

Cited by 9 publications

(11 citation statements)

References 20 publications

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“…A virtual motivic version of topological mirror symmetry has already been established by Loeser and Wyss [48] who prove an equality of (orbifold) virtual Chow motives in the Grothendieck ring of Chow motives using motivic integration and the ideas of [32]. Our result implies and can be thought of as a categorification of the main theorem in [48], see Corollary 6.20 (i).…”

supporting

confidence: 62%

Motivic mirror symmetry for Higgs bundles

Hoskins¹,

Lehalleur²

2022

Preprint

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We prove that the (orbifold) motives of the moduli spaces of twisted SL n and PGL n -Higgs bundles of coprime rank and degree on a smooth projective curve over an algebraically closed field of characteristic zero are isomorphic in Voevodsky's triangulated category of motives. The equality of (orbifold) Hodge numbers of these moduli spaces was conjectured by Hausel and Thaddeus and recently proven by Groechenig, Ziegler and Wyss via p-adic integration and then by Maulik and Shen using the decomposition theorem, an analysis of the supports of twisted Hitchin fibrations and vanishing cycles. Our proof combines the geometric ideas of Maulik and Shen with the conservativity of the Betti realisation on abelian motives; to apply the latter, we prove that the relevant motives are abelian. In particular, we prove that the motive of the twisted SL n -Higgs moduli space is abelian, building on our previous work in the GL n -case. 2 VICTORIA HOSKINS AND SIMON PEPIN LEHALLEUR B.1. A formula for the motive of the stack of bundles with fixed determinant 37 B.2. Relative formulae for families of curves 39 Appendix C. Dimension formulae 41 References 41

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confidence: 62%

Motivic mirror symmetry for Higgs bundles

Hoskins¹,

Lehalleur²

2022

Preprint

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“…A refined version of the Hausel–Thaddeus conjecture was previously proven by Gröchenig, Wyss and Ziegler [17, Theorem 7.24] by p -adic integration, and was generalised by Loeser and Wyss [29, Remark 5.3.4] by motivic integration. Note that our refined version in equation (7) is slightly different from the versions of [17, 29], since the right-hand side of equation (7) depends on the degree of L , whereas the corresponding term in [17, 29] is independent of this degree. Instead, our refined version is closer to the conjectures formulated by Hausel [20, Conjectures 4.5 and 5.9].…”

Section: The Hausel–thaddeus Conjecturementioning

confidence: 81%

“…The following theorem is a direct consequence of Theorem 0.3 which proves the Hausel–Thaddeus conjecture and a refinement of it; see [29] for an explanation of how the right-hand side is equivalent to the gerby description just given:…”

Section: The Hausel–thaddeus Conjecturementioning

confidence: 93%

Endoscopic decompositions and the Hausel–Thaddeus conjecture

Maulik

Shen

2021

Forum of Mathematics, Pi

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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.

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“…This conjecture is stated in terms of stringy Hodge numbers twisted by µ ngerbes. Later, this result was refined and proved at the motivic level by Loeser-Wyss [LW21]. It would be interesting to incorporate wild actions into research in this direction.…”

Section: Generalization In Various Directionsmentioning

confidence: 91%

Open problems in the wild McKay correspondence and related fields

Yasuda¹

2021

Preprint

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The wild McKay correspondence is a form of McKay correspondence in terms of stringy invariants that is generalized to arbitrary characteristics. It gives rise to an interesting connection between the geometry of wild quotient varieties and arithmetic on extensions of local fields. The principal purpose of this article is to collect open problems on the wild McKay correspondence, as well as those in related fields that the author believes are interesting or important. It also serves as a survey on the present state of these fields. Contents 1. Introduction Convention 2. Brief guide to the wild McKay correspondence and stringy motives 2.1. Wild actions and wild quotients 2.2. Stringy motives 2.3. The wild McKay correspondence 3. Log-terminal singularities 4. Crepant resolutions 5. Rationality 6. Duality 7. The v-function/Fröhlich's module resolvent 8. The moduli space ∆ G 9. Non-linear actions 10. Generalization in various directions 11. Relation to other theories on wild ramification 12. The derived wild McKay correspondence 13. The McKay correspondence over global fields 14. Stringy motives and quasi-étale Galois coverings References

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Motivic integration on the Hitchin fibration

Cited by 9 publications

References 20 publications

Motivic mirror symmetry for Higgs bundles

Motivic mirror symmetry for Higgs bundles

Endoscopic decompositions and the Hausel–Thaddeus conjecture

Open problems in the wild McKay correspondence and related fields

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