# Crystallinity properties of complex rigid local systems [not online]

*Helene Esnault (Freie Universität Berlin)*

**Fri Apr 28, 19:00-20:00 (4 weeks ago)**

**Abstract: **Joint work in progress with Michael Groechenig

We prove in all generality that on a smooth complex quasi-projective variety $X$, Rigid connections yield $F$-isocrystals on almost all good reductions $X_{\mathbb F_q}$ and that rigid local systems yield crystalline local systems on $X_K$ for $K$ the field of fractions of the Witt vectors of a finite field $\mathbb F_q$, for almost all $X_{\mathbb F_q}$. This improves our earlier work where, if $X$ was not projective, we assumed a strong cohomological condition (which is fulfilled for Shimura varieties of real rank $\geq 2$), and we obtained only infinitely many $\mathbb F_q$ of growing characteristic. While the earlier proof was via characteristic $p$, the new one is purely $p$-adic and uses $p$-adic topology.

We shall discuss the projective case during the lecture.

algebraic geometry

Audience: researchers in the topic

**Stanford algebraic geometry seminar **

**Series comments: **The seminar was online for a significant period of time, but for now is solely in person.
More seminar information (including slides and videos, when available): agstanford.com

Organizer: | Ravi Vakil* |

*contact for this listing |