Speaker: Prof. Hélène Esnault
Time: 16:30, May 29, 2025
Venue: Room 612, A6, Institute of Mathematics-VAST
Online (Join Zoom Meeting) tại link: https://zoom.us/j/99636681387?pwd=0WscBnehOJig68SqctGluVuA3RwraE.1
Abstract: If $X$ is a smooth projective variety defined over the field of complex numbers, its $i$-th Betti cohomology $H^i(X, mathbb C)$ is said to have {it coniveau one} if there is a Zariski dense open $Usubset X$ such that the restriction map $H^i(X,mathbb C) to H^i(U,mathbb C)$ dies. Equivalently, the restriction map to the generic point of $X$ in $i$-th cohomology vanishes. Grothendieck’s generalized Hodge conjecture is in general difficult to express as one needs the notion of Hodge sub-structure, but one particular instance has a purely algebraic formulation. It predicts that if $X$ has no non-trivial global differential forms of degree $i$, then $H^i(X, mathbb C)$ should have coniveau one. The converse is easily seen to be true. Aside of $i=1,2$, for which complex Hodge theory gives a positive answer, we know nothing. On the other hand, the philosophy behind is very useful to draw analogies, e.g. it helps to find rational points over finite fields of rationally connected varieties (Lang-Manin conjecture). So it is worth to try to understand whether more modern $p$-adic methods yield some non-trivial information.
With {bf Mark Kisin and Alexander Petrov}, in {bf work in progress}, we formulate and prove a vanishing result in the separate quotient of $p$-completed de Rham cohomology, and a weaker version in the separate quotient of prismatic cohomology. I’ll present the ‘program’ and a few questions which at present are not understood. |
