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.
Abstract. We study relative zero cycles on the universal polarized K3 surface X → Fg of degree 2g − 2. It was asked by O'Grady if the restriction of any class in CH 2 (X) to a closed fiber Xs is a multiple of the Beauville-Voisin canonical class c Xs ∈ CH 0 (Xs). Using Mukai models, we give an affirmative answer to this question for g ≤ 10 and g = 12, 13, 16, 18, 20. 0. Introduction Throughout, we work over the complex numbers. Let S be a projective K3 surface. In [2], Beauville and Voisin studied the Chow ring CH * (S) of S. They showed that there is a canonical class c S ∈ CH 0 (S) represented by a point on a rational curve in S, which satisfies the following properties:(i) The intersection of two divisor classes on S always lies in Zc S ⊂ CH 0 (S).(ii) The second Chern class c 2 (T S ) equals 24c S ∈ CH 0 (S). This result is rather surprising since the Chow group CH 0 (S) is infinite-dimensional by Mumford's theorem [7].Let F g denote the moduli space of (primitively) polarized K3 surfaces of degree 2g − 2. For g ≥ 3, let F 0 g ⊂ F g be the open dense subset parametrizing polarized K3 surfaces with trivial automorphism groups, which carries a universal family X → F 0 g . Motivated by Franchetta's conjecture on the moduli spaces of curves (see [1]), O'Grady asked the following question in [12], referred to as the generalized Franchetta conjecture.Question 0.1 (Generalized Franchetta conjecture). Given a class α ∈ CH 2 (X) and a closed point s ∈ F 0 g , is it true that α| Xs ∈ Zc Xs ? The goal of this paper is to give an affirmative answer to Question 0.1 for a list of small values of g. By the work of Mukai [8,9,10,11], for these g a general polarized K3 surface can be realized in a variety with "small" Chow groups as a complete intersection with respect to a vector bundle.Theorem 0.2. The generalized Franchetta conjecture holds for g ≤ 10 and g = 12, 13, 16, 18, 20.The paper is organized as follows. In Section 1 we review Mukai's constructions and make some comments about Question 0.1. In Section 2 we prove Theorem 0.2 for all cases except g = 13, 16. Two independent proofs are presented, one using Voisin's result [17], the other via a direct calculation. The cases g = 13, 16 have a different flavor and are treated in Section 3. Acknowledgement. We are grateful to Rahul Pandharipande for his constant support and his enthusiasm in this project. We also thank Kieran O'Grady for his careful reading of a preliminary version of this paper.
We prove that the intersection cohomology (together with the perverse and the Hodge filtrations) for the moduli space of one-dimensional semistable sheaves supported in an ample curve class on a toric del Pezzo surface is independent of the Euler characteristic of the sheaves. We also prove an analogous result for the moduli space of semistable Higgs bundles with respect to an effective divisor D of degree deg(D) > 2g − 2. Our results confirm the cohomological χ-independence conjecture by Bousseau for P 2 , and verify Toda's conjecture for Gopakumar-Vafa invariants for certain local curves and local surfaces.For the proof, we combine a generalized version of Ngô's support theorem, a dimension estimate for the stacky Hilbert-Chow morphism, and a splitting theorem for the morphism from the moduli stack to the good GIT quotient.
We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus 2 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.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.
We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialization, recover the topological mirror symmetry conjecture of Hausel-Thaddeus concerning SLn-and PGLn-Higgs bundles. This provides a complete description of the cohomology of the moduli space of stable SLn-Higgs bundles in terms of the tautological classes, and gives a new proof of the Hausel-Thaddeus conjecture, proven recently by Gröchenig-Wyss-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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