Status  Confirmed 
Seminar Series  MATHIHES 
Subjects  math 
Date  Thursday 20 October 2022 
Time  11:00 
Institute  IHES 
Seminar Room  Amphithéâtre Léon Motchane 
Speaker's Last Name  Kubrak 
Speaker's First Name  Dmitry 
Speaker's Email Address  
Speaker's Institution  IHES 
Title  Hodgeproper Stacks and Totaro's Conjecture, part 1 
Abstract  I will talk about a series of works with Artem Prikhodko where we develop a version of padic Hodge theory in the setting of Artin stacks. One of the main motivations for our project was a conjecture by Totaro: namely, based on his concrete computations, he suggested that the dimension of mod p de Rham cohomology of the classifying stack BG for G reductive might always be bounded from below by the dimension of the Fpsingular cohomology on the classifying space BG(C) of the Lie group G(C) of complex points of G. For smooth and proper schemes such an inequality is a consequence of integral padic Hodge theory in the form proved by BhattMorrowScholze; however, their results can not be applied here directly since BG is not proper. To prove Totaro's conjecture, using the theory of prismatic cohomology, we develop integral padic Hodge theory in a more general setting of Hodgeproper stacks: these are stacks that only look proper from the point of view of its Hodge cohomology. However, one problem then still remains: namely, the étale comparison we get is with the étale cohomology of the Raynaud generic fiber, which a priori agrees with the algebraic generic fiber (and then complex points) only in the smooth proper setting. Nevertheless, we prove that the two étale cohomology theories still agree at least for quotient stacks [X/G] with X smooth and proper and G reductive. This then implies Totaro's conjecture by plugging X=pt. In further work we also show that after inverting p the two étale cohomology agree for any Hodgeproper stack, which sets up rational padic Hodge theory (with the crystalline and de Rham comparisons, and HodgeTate decomposition) in this setting. If time permits, I will also tell about some explicit computations of cohomology of reductive groups in characteristic p that one can perform using the above comparison in the case of BG (this is a series of joint works with Federico Scavia and Anlong Chua). 
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