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1. Deformations and moduli of irregular canonical covers with $$K^2=4p_g-8$$

   https://doi.org/10.1007/s13163-023-00462-5  

   Bangere, Purnaprajna ; Gallego, Francisco Javier ; Mukherjee, Jayan ; Raychaudhury, Debaditya ( March 2023 , Revista Matemática Complutense)

   In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying$$K^2 = 4p_g-8$$$K^2=4p_g-8$, for any even integer$$p_g\ge 4$$$p_g\ge 4$. These surfaces also have unbounded irregularityq. We carry out our study by investigating the deformations of the canonical morphism$$\varphi :X\rightarrow {\mathbb {P}}^N$$$\phi :X\to P^N$, where$$\varphi $$$\phi$is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):5489-5507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of$$\varphi $$$\phi$factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of$$\varphi $$$\phi$is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations ofXare unobstructed even though$$H^2(T_X)$$$H^2\left(T_X\right)$does not vanish. Consequently,Xbelongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality$$p_g > 2q-4$$$p_g>2q-4$, with an irreducible component that has a proper quadruple sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with$$K^2 = 2p_g- 4$$$K^2=2p_g-4$, studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus$$g\ge 3$$$g\ge 3$, for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map.

    

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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. Open 𝑟-Spin Theory I: Foundations

   https://doi.org/10.1093/imrn/rnaa345  

   Buryak, Alexandr ; Clader, Emily ; Tessler, Ran J ( February 2021 , International Mathematics Research Notices)

   null (Ed.)

   Abstract We lay the foundation for a version of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define the notion of $r$-spin disks, their moduli space, and the Witten bundle; we show that the moduli space is a compact smooth orientable orbifold with corners, and we prove that the Witten bundle is canonically relatively oriented relative to the moduli space. In the sequel to this paper, we use these constructions to define open $r$-spin intersection theory and relate it to the Gelfand–Dickey hierarchy, thus providing an analog of Witten’s $r$-spin conjecture in the open setting. 

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4. Largest hyperbolic actions and quasi-parabolic actions in groups

   https://doi.org/10.1142/S1793525322500066  

   Abbott, Carolyn R. ; Rasmussen, Alexander J. ( January 2022 , Journal of Topology and Analysis)

   The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. The resulting poset thus gives rise to a notion of the “best” hyperbolic action of a group as the largest element of this poset, if such an element exists. We call such an action a largest hyperbolic action. While hyperbolic groups admit the largest hyperbolic actions, we give evidence in this paper that this phenomenon is rare for non-hyperbolic groups. In particular, we prove that many families of groups of geometric origin do not have the largest hyperbolic actions, including for instance many 3-manifold groups and most mapping class groups. Our proofs use the quasi-trees of metric spaces of Bestvina–Bromberg–Fujiwara, among other tools. In addition, we give a complete characterization of the poset of hyperbolic actions of Anosov mapping torus groups, and we show that mapping class groups of closed surfaces of genus at least two have hyperbolic actions which are comparable only to the trivial action. 

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5. Hyperbolic families and coloring graphs on surfaces

   https://doi.org/https://doi.org/10.1090/btran/26  

   Postle, Luke ; Thomas, Robin ( December 2018 , Transactions of the American Mathematical Society. Series B)

   We develop a theory of linear isoperimetric inequalities for graphs on surfaces and apply it to coloring problems, as follows. Let $ G$ be a graph embedded in a fixed surface $ \Sigma $ of genus $ g$ and let $ L=(L(v):v\in V(G))$ be a collection of lists such that either each list has size at least five, or each list has size at least four and $ G$ is triangle-free, or each list has size at least three and $ G$ has no cycle of length four or less. An $ L$-coloring of $ G$ is a mapping $ \phi $ with domain $ V(G)$ such that $ \phi (v)\in L(v)$ for every $ v\in V(G)$ and $ \phi (v)\ne \phi (u)$ for every pair of adjacent vertices $ u,v\in V(G)$. We prove if every non-null-homotopic cycle in $ G$ has length $ \Omega (\log g)$, then $ G$ has an $ L$-coloring, if $ G$ does not have an $ L$-coloring, but every proper subgraph does (``$ L$-critical graph''), then $ \vert V(G)\vert=O(g)$, if every non-null-homotopic cycle in $ G$ has length $ \Omega (g)$, and a set $ X\subseteq V(G)$ of vertices that are pairwise at distance $ \Omega (1)$ is precolored from the corresponding lists, then the precoloring extends to an $ L$-coloring of $ G$, if every non-null-homotopic cycle in $ G$ has length $ \Omega (g)$, and the graph $ G$ is allowed to have crossings, but every two crossings are at distance $ \Omega (1)$, then $ G$ has an $ L$-coloring, if $ G$ has at least one $ L$-coloring, then it has at least $ 2^{\Omega (\vert V(G)\vert)}$ distinct $ L$-colorings. We show that the above assertions are consequences of certain isoperimetric inequalities satisfied by $ L$-critical graphs, and we study the structure of families of embedded graphs that satisfy those inequalities. It follows that the above assertions hold for other coloring problems, as long as the corresponding critical graphs satisfy the same inequalities. 

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