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1. 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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2. Projective Compactification of Dolbeault Moduli Spaces

   https://doi.org/10.1093/imrn/rnaa069  

   de Cataldo, Mark Andrea ( April 2020 , International Mathematics Research Notices)

   We construct a relative projective compactification of Dolbeault moduli spaces of Higgs bundles for reductive algebraic groups on families of projective manifolds that is compatible with the Hitchin morphism.

    

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3. Endoscopic decompositions and the Hausel–Thaddeus conjecture

   https://doi.org/10.1017/fmp.2021.7  

   Maulik, Davesh ; Shen, Junliang ( January 2021 , Forum of Mathematics, Pi)

   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. 

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4. Moduli Spaces of Generalized Hyperpolygons

   https://doi.org/10.1093/qmath/haaa036  

   Rayan, Steven ; Schaposnik, Laura P ( December 2020 , The Quarterly Journal of Mathematics)

   null (Ed.)

   Abstract We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such data, we associate an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet, thereby embedding the Nakajima quiver variety into a Hitchin system on a punctured genus-g Riemann surface (generally with positive codimension). We show that, under certain assumptions on flag types, the space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system of Gelfand–Tsetlin type, inherited from the reduction of partial flag varieties. In the case where all flags are complete, we present the Hamiltonians explicitly. We also remark upon the discretization of the Hitchin equations given by hyperpolygons, the construction of triple branes (in the sense of Kapustin–Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents). 

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5. 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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