Speaker: Prof. Anna Cadoret (Sorbonne Université)

Time & Venue: 15:00-16:00  MCM110
Title: Variation of Tannaka group of perverse sheaves over abelian varieties in families
Abstract: Given a relative perverse sheaf P (in the sense of Hansen-Scholze) over an abelian scheme A/S over a smooth irreducible variety S, whose restriction to the geometric generic fiber A_\eta is semisimple (in the corresponding Tannakian category), we prove that the loci of all s in S where the connected component of the derived subgroup of the Tannaka group of P restricted to A_s degenerates is not Zariski-dense. This builds on and enhances previous works by Kramer and Weissauer. Applying our result to intersection complexes, and using computations by Javanpeykar-Lehn-Kramer-Maculan, we derive geometric applications. This is a joint work with Haohao Liu.

Speaker: Prof. Tomoyuki Abe (The University of Tokyo)

Time & Venue: 16:30-17:30  MCM110
Title: Serre's conjecture on Artin character in the geometric case
Abstract: Given a finite group G which acts on a regular local ring A with some conditions, Serre defined an integer-valued function on G. When A is a discrete valuation ring, this coincides with the classical Artin character, and as a consequence, it is the associated character of the Artin representation. He conjectured that the same picture holds in the higher dimensional case: the function is an associated character of an l-adic representation. When A is equal characteristic, I want to show how this conjecture can be proven as a small application of Lu-Zheng's categorical trace for etale sheaves.