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1. THE INTEGRAL COHOMOLOGY OF THE HILBERT SCHEME OF TWO POINTS

   https://doi.org/10.1017/fms.2016.5  

   TOTARO, BURT (January 2016, Forum of Mathematics, Sigma)

   null (Ed.)

   The Hilbert scheme $$X^{[a]}$$ of points on a complex manifold $$X$$ is a compactification of the configuration space of $$a$$ -element subsets of $$X$$ . The integral cohomology of $$X^{[a]}$$ is more subtle than the rational cohomology. In this paper, we compute the mod 2 cohomology of $$X^{[2]}$$ for any complex manifold $$X$$ , and the integral cohomology of $$X^{[2]}$$ when $$X$$ has torsion-free cohomology. 

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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. On birational boundedness of foliated surfaces

   https://doi.org/10.1515/crelle-2020-0009  

   Hacon, Christopher D.; Langer, Adrian (January 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract In this paper we prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function {P:\mathbb{Z}_{\geq 0}\to\mathbb{Z}} , then there exists an integer {N>0} such that if {(X,{\mathcal{F}})} is a canonical or nef model of a foliation of general type with Hilbert polynomial {\chi(X,{\mathcal{O}}_{X}(mK_{\mathcal{F}}))=P(m)} for all {m\in\mathbb{Z}_{\geq 0}} , then {|mK_{\mathcal{F}}|} defines a birational map for all {m\geq N} . On the way, we also prove a Grauert–Riemenschneider-type vanishing theorem for foliated surfaces with canonical singularities. 

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4. Flat line bundles and the Cappell-Miller torsion in Arakelov geometry

   https://doi.org/10.24033/asens.2409  

   Freixas i Montplet, Gerard; Wentworth, Richard A. (January 2019, Annales scientifiques de lÉcole normale supérieure)

   null (Ed.)

   In this paper, we extend Deligne’s functorial Riemann-Roch isomorphism for Hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and ?-product of Gillet-Soulé are replaced with complex valued logarithms. On the determinant of cohomology side, we show that the Cappell-Miller torsion is the appropriate counterpart of the Quillen metric. On the Deligne pairing side, the logarithm is a refinement of the intersection connections considered in a previous work. The construction naturally leads to an Arakelov theory for flat line bundles on arithmetic surfaces and produces arithmetic intersection numbers valued in C/πi Z. In this context we prove an arithmetic Riemann-Roch theorem. This realizes a program proposed by Cappell-Miller to show that their holomorphic torsion exhibits properties similar to those of the Quillen metric proved by Bismut, Gillet and Soulé. Finally, we give examples that clarify the kind of invariants that the formalism captures; namely, periods of differential forms. 

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5. Automorphic cohomology, motivic cohomology, and the adjoint L-function

   Prasanna, Kartik A.; Venkatesh, Akshay (July 2021, Asterisque)

   We propose a relationship between the cohomology of arithmetic groups, and the motivic cohomology of certain (Langlands-)attached motives. The motivic cohomology group in question is that related, by Beilinson’s conjecture, to the adjoint L-function at s=1. We present evidence for the conjecture using the theory of periods of automorphic forms, and using analytic torsion. 

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