<i>P</i> = <i>W</i> for Lagrangian fibrations and degenerations of hyper-Kähler manifolds

Harder, Andrew; Li, Zhiyuan; Shen, Junliang; Yin, Qizheng

doi:10.1017/fms.2021.31

Cited by 11 publications

(4 citation statements)

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“…A certain limit of the conjecture appears to be related to a comparison of limiting behavior of the Hitchin fibration with the geometry of the boundary complex of the character variety [S5]. Relationships between perverse and weight filtrations have also been found in other settings of hyperkähler geometry [dCHM2,Har,HLSY]. A similar sounding (but at present not directly related) statement has been found in homological mirror symmetry [HKP].…”

Section: Introductionmentioning

confidence: 85%

Deletion-contraction triangles for Hausel-Proudfoot varieties

Dancso¹,

McBreen²,

Shende³

2019

Preprint

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To a graph, Hausel and Proudfoot associate two complex manifolds, B and D, which behave, respectively like moduli of local systems on a Riemann surface, and moduli of Higgs bundles. For instance, B is a moduli space of microlocal sheaves, which generalize local systems, and D carries the structure of a complex integrable system.We show the Euler characteristics of these varieties count spanning subtrees of the graph, and the point-count over a finite field for B is a generating polynomial for spanning subgraphs. This polynomial satisfies a deletion-contraction relation, which we lift to a deletion-contraction exact triangle for the cohomology of B. There is a corresponding triangle for D.Finally, we prove B and D are diffeomorphic, that the diffeomorphism carries the weight filtration on the cohomology of B to the perverse Leray filtration on the cohomology of D, and that all these structures are compatible with the deletion-contraction triangles.

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Section: Introductionmentioning

confidence: 85%

Deletion-contraction triangles for Hausel-Proudfoot varieties

Dancso¹,

McBreen²,

Shende³

2019

Preprint

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“…In fact as we will see in its proof, (11) is equivalent to (8). Hence Conjecture 0.1 can be viewed as extending the perverse-Hodge symmetry from the 0-section B to a neighborhood B.…”

mentioning

confidence: 89%

“…The perverse-Hodge symmetry. The motivation for Conjecture 0.1 is an effort to understand and categorify the perverse-Hodge symmetry for Lagrangian fibrations [31]; see also [11,10,16,32].…”

mentioning

confidence: 99%

Fourier-Mukai transforms and the decomposition theorem for integrable systems

Maulik¹,

Shen²,

Yin³

2023

Preprint

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We study the interplay between the Fourier-Mukai transform and the decomposition theorem for an integrable system π : M → B. Our main conjecture is that the Fourier-Mukai transform of sheaves of Kähler differentials, after restriction to the formal neighborhood of the zero section, are quantized by the Hodge modules arising in the decomposition theorem for π. For an integrable system, our formulation unifies the Fourier-Mukai calculation of the structure sheaf by Arinkin-Fedorov, the theorem of the higher direct images by Matsushita, and the "perverse = Hodge" identity by the second and the third authors.As evidence, we show that these Fourier-Mukai images are Cohen-Macaulay sheaves with middle-dimensional support on the relative Picard space, with support governed by the higher discriminants of the integrable system. We also prove the conjecture for smooth integrable systems and certain 2-dimensional families with nodal singular fibers. Finally, we sketch the proof when cuspidal fibers appear.

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“…This conjecture was originally made for Hitchin systems, and proved for

\mathrm{GL}(2)

‐,

\mathrm{SL}(2)

‐, and

\mathrm{PGL}(2)

‐Hitchin systems on curves of arbitrary genus [6]. A numerical version of the conjecture was proved for Lagrangian fibrations on smooth compact holomorphic symplectic manifolds by (Junliang) Shen and Yin [25] (for a related result in the compact case, see also Harder, Li, Shen, and Yin [13]). Subsequently, de Cataldo, Maulik, and Shen [7] proved the conjecture for

\mathrm{GL}(n)

‐Hitchin systems on genus two curves by using an analogue of Donagi, Ein, and Lazarsfeld's deformation coming from the deformation of an abelian surface to the normal cone of an embedded curve.…”

Section: Introductionmentioning

confidence: 99%