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1. Motivic Action on Coherent Cohomology of Hilbert Modular Varieties

   https://doi.org/10.1093/imrn/rnac126  

   Horawa, Aleksander (May 2022, International Mathematics Research Notices)

   Abstract We propose an action of a certain motivic cohomology group on the coherent cohomology of Hilbert modular varieties, extending conjectures of Venkatesh, Prasanna, and Harris. The action is described in two ways: on cohomology modulo $$p$$ and over $${\mathbb {C}}$$, and we conjecture that they both lift to an action on cohomology with integral coefficients. The conjecture is supported by theoretical evidence based on Stark’s conjecture on special values of Artin $$L$$-functions and by numerical evidence in base change cases. 

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2. $$L^{2}$$-Cohomology of quasi-fibered boundary metrics

   https://doi.org/10.1007/s00222-024-01253-5  

   Kottke, Chris; Rochon, Frédéric (June 2024, Inventiones mathematicae)

   We develop new techniques to compute the weighted L2-cohomology of quasi- fibered boundary metrics (QFB-metrics). Combined with the decay of L2-harmonic forms obtained in a companion paper, this allows us to compute the reduced L2- cohomology for various classes of QFB-metrics. Our results applies in particular to the Nakajima metric on the Hilbert scheme of n points on C2, for which we can show that the Vafa-Witten conjecture holds. Using the compactification of the monopole moduli space announced by Fritzsch, the first author and Singer, we can also give a proof of the Sen conjecture for the monopole moduli space of magnetic charge 3. 

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3. Ranks of matrix factorizations and sheaf cohomology

   https://doi.org/10.1090/proc/17257  

   Brown, Michael; Walker, Mark (August 2025, Proceedings of the American Mathematical Society)

   Buchweitz-Greuel-Schreyer conjectured in 1987 a lower bound on the ranks of matrix factorizations over certain local hypersurface rings [Invent. Math. 88 (1987), pp. 165–182]. We study a graded version of this conjecture, and we show that it implies a novel conjecture concerning the cohomology of sheaves over non-Fano projective hypersurfaces. 

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4. Essential dimension via prismatic cohomology

   https://doi.org/10.1215/00127094-2023-0071  

   Farb, Benson; Kisin, Mark; Wolfson, Jesse (October 2024, Duke Mathematical Journal)

   NA (Ed.)

   For X a smooth, proper complex variety we show that for p≫0, the restriction of the mod p cohomology Hi(X,Fp) to any Zariski open has dimension at least h0,iX . The proof uses the prismatic cohomology of Bhatt and Scholze. We use this result to obtain lower bounds on the p-essential dimension of covers of complex varieties. For example, we prove the p-incompressibility of the mod p homology cover of an abelian variety, confirming a conjecture of Brosnan for sufficiently large p. By combining these techniques with the theory of toroidal compactifications of Shimura varieties, we show that for any Hermitian symmetric domain X, there exist p-congruence covers that are p-incompressible. 

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5. On the codimension-two cohomology of SLn(ℤ).

   https://doi.org/10.1016/j.aim.2024.109795  

   Brück, Benjamin; Miller, Jeremy; Patzt, Peter; Sroka, Robin J; Wilson, Jennifer CH (August 2024, Advances in Mathematics)

   Borel–Serre proved that SL_n(Z) is a virtual duality group of dimension (n choose 2) and the Steinberg module St_n(Q) is its dualizing module. This module is the top-dimensional homology group of the Tits building associated to SL_n(Q). We determine the “relations among the relations” of this Steinberg module. That is, we construct an explicit partial resolution of length two of the SL_n(Z)-module St_n(Q). We use this partial resolution to show the codimension-2 rational cohomology group of SLn(Z) vanishes for n ≥ 3. This resolves a case of a conjecture of Church–Farb–Putman. We also produce lower bounds for the codimension-1 cohomology of certain congruence subgroups of SLn(Z). 

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