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1. Perverse Leray filtration and specialisation with applications to the Hitchin morphism

   https://doi.org/10.1017/S0305004121000293  

   DE CATALDO, MARK ANDREA (March 2022, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We initiate and develop a framework to handle the specialisation morphism as a filtered morphism for the perverse, and for the perverse Leray filtration, on the cohomology with constructible coefficients of varieties and morphisms parameterised by a curve. As an application, we use this framework to carry out a detailed study of filtered specialisation for the Hitchin morphisms associated with the compactification of Dolbeault moduli spaces in [de 2018]. 

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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. Wonderful compactifications and rational curves with cyclic action

   https://doi.org/10.1017/fms.2023.26  

   Clader, Emily; Damiolini, Chiara; Li, Shiyue; Ramadas, Rohini (January 2023, Forum of Mathematics, Sigma)

   Abstract We prove that the moduli space of rational curves with cyclic action, constructed in our previous work, is realizable as a wonderful compactification of the complement of a hyperplane arrangement in a product of projective spaces. By proving a general result on such wonderful compactifications, we conclude that this moduli space is Chow-equivalent to an explicit toric variety (whose fan can be understood as a tropical version of the moduli space), from which a computation of its Chow ring follows. 

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4. Billiards and Teichmüller curves

   https://doi.org/10.1090/bull/1782  

   McMullen, Curtis (April 2023, Bulletin of the American Mathematical Society)

   A Teichmüller curve V ⊂ M g V \subset \mathcal {M}_g is an isometrically immersed algebraic curve in the moduli space of Riemann surfaces. These rare, extremal objects are related to billiards in polygons, Hodge theory, algebraic geometry and surface topology. This paper presents the six known families of primitive Teichmüller curves that have been discovered over the past 30 years, and a selection of open problems. 

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5. A polyfold proof of the Arnold conjecture

   https://doi.org/10.1007/s00029-021-00680-z  

   Filippenko, Benjamin; Wehrheim, Katrin (February 2022, Selecta Mathematica)

   Abstract We give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds M via a direct Piunikhin–Salamon–Schwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from $${{\mathbb {C}}{\mathbb {P}}}^1\times M$$ C P 1 × M to $${{\mathbb {C}}}^+ \times M$$ C + × M and $${{\mathbb {C}}}^-\times M$$ C - × M , as developed by Fish–Hofer–Wysocki–Zehnder as part of the Symplectic Field Theory package. To make the paper self-contained we include all polyfold assumptions, describe the coherent perturbation iteration in detail, and prove an abstract regularization theorem for moduli spaces with evaluation maps relative to a countable collection of submanifolds. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish. 

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