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1. A support theorem for\break the Hitchin fibration: The case of GL n and K C

   https://doi.org/10.1515/crelle-2021-0045  

   de Cataldo, Mark Andrea ; Heinloth, Jochen ; Migliorini, Luca ( August 2021 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   We compute the supports of the perverse cohomology sheaves of the Hitchin fibration for${\mathrm{GL}}_n${\mathrm{GL}_{n}}over the locus of reduced spectral curves. In contrast to the case of meromorphic Higgs fields we find additional supports at the loci of reducible spectral curves. Their contribution to the global cohomology is governed by a finite twist of Hitchin fibrations for Levi subgroups. The corresponding summands give non-trivial contributions to the cohomology of the moduli spaces for every$n\ge 2${n\geq{2}}. A key ingredient is a restriction result for intersection cohomology sheaves that allows us to compare the fibration to the one defined over versal deformations of spectral curves.

    

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2. Hitchin fibrations, abelian surfaces, and the P=W conjecture

   https://doi.org/10.1090/jams/989  

   de Cataldo, Mark ; Maulik, Davesh ; Shen, Junliang ( November 2021 , Journal of the American Mathematical Society)

   We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus 2 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. 

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3. Geometry of the logarithmic Hodge moduli space

   https://doi.org/10.1112/jlms.12857  

   de Cataldo, Mark Andrea ; Herrero, Andres Fernandez ; Zhang, Siqing ( January 2024 , Journal of the London Mathematical Society)

   We show the smoothness over the affine line of the Hodge moduli space of logarithmic ‐connections of coprime rank and degree on a smooth projective curve with geometrically integral fibers over an arbitrary Noetherian base. When the base is a field, we also prove that the Hodge moduli space is geometrically integral. Along the way, we prove the same results for the corresponding moduli spaces of logarithmic Higgs bundles and of logarithmic connections. We use smoothness to derive specialization isomorphisms on the étale cohomology rings of these moduli spaces; this includes the special case when the base is of mixed characteristic. In the special case where the base is a separably closed field of positive characteristic, we show that these isomorphisms are filtered isomorphisms for the perverse filtrations associated with the corresponding Hitchin‐type morphisms.

    

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4. Rigidity properties of the cotangent complex

   https://doi.org/10.1090/jams/1000  

   Briggs, Benjamin ; Iyengar, Srikanth ( January 2023 , Journal of the American Mathematical Society)

   This work concerns a map φ : R → S \varphi \colon R\to S of commutative noetherian rings, locally of finite flat dimension. It is proved that the André-Quillen homology functors are rigid, namely, if D n ( S / R ; − ) = 0 \mathrm {D}_n(S/R;-)=0 for some n ≥ 1 n\ge 1 , then D i ( S / R ; − ) = 0 \mathrm {D}_i(S/R;-)=0 for all i ≥ 2 i\ge 2 and φ {\varphi } is locally complete intersection. This extends Avramov’s theorem that draws the same conclusion assuming D n ( S / R ; − ) \mathrm {D}_n(S/R;-) vanishes for all n ≫ 0 n\gg 0 , confirming a conjecture of Quillen. The rigidity of André-Quillen functors is deduced from a more general result about the higher cotangent modules which answers a question raised by Avramov and Herzog, and subsumes a conjecture of Vasconcelos that was proved recently by the first author. The new insight leading to these results concerns the equivariance of a map from André-Quillen cohomology to Hochschild cohomology defined using the universal Atiyah class of φ \varphi . 

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5. Spectral Data for Spin Higgs Bundles

   https://doi.org/10.1093/imrn/rny296  

   Mukhopadhyay, Swarnava ; Wentworth, Richard ( January 2019 , International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper we determine the spectral data parametrizing Higgs bundles in a generic fiber of the Hitchin map for the case where the structure group is the special Clifford group with fixed Clifford norm. These are spin and “twisted” spin Higgs bundles. The method used relates variations in spectral data with respect to the Hecke transformations for orthogonal bundles introduced by Abe. The explicit description also recovers a result from the geometric Langlands program, which states that the fibers of the Hitchin map are the dual abelian varieties to the corresponding fibers of the moduli spaces of projective orthogonal Higgs bundles (in the even case) and projective symplectic Higgs bundles (in the odd case). 

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