##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## Adrian Vasiu

#### SUNY

## Cohomological Invariants of Projective Varieties in Positive Characteristic

##### Abstract:

Let X be a projective smooth variety over an algebraically closed field k. If k has characteristic zero, then the singular (Betti) cohomology groups of X are finitely generated abelian groups and therefore all the invariants associated to them are discrete and in fact do not change under good deformations. If k has positive characteristic, then the crystalline cohomology groups of X have a much richer structure and are called F-crystals over k. In particular, one can associate to them many subtle invariants which vary a lot under good deformations and which could be of either discrete or continuous nature. We present an accessible survey of the classification of F-crystals over k via subtle invariants with an emphasis on the recent results obtain by us, by Gabber and us, and by Lau, Nicole, and us.

Cristian Popescu

### February 21, 2013

### 3:00 PM

### AP&M 6402

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