Geometry Seminar

 

University of Tor Vergata, Department of Mathematics

6th of May 2025, 14:30-16:00,room D’Antoni

 

 

 

 

 

 

On the Euler characteristic of ordinary irregular varieties in positive characteristic

 

Jefferson Jacques Christophe Baudin


École polytechnique fédérale de Lausanne (EPFL)

 

 

 

 

Over the complex numbers, generic vanishing theory is useful for studying the geometry of irregular varieties. A standard application of this theory is that if X is a smooth, complex projective variety of maximal Albanese dimension (i.e. dim(alb(X)) = dim(X)), then the Euler characteristic of the sheaf of top forms is non-negative. This relies on vanishing theorems of analytic nature.

In this talk, we will show that the same statement holds in positive characteristic, assuming further that the Frobenius morphism acts bijectively on the cohomology (such a hypothesis tends to be true for "most" varieties). If we also assume that our variety is not of general type, then we also show that its Euler characteristic is in fact zero, and that the image of the Albanese morphism is fibered by abelian varieties (which are well-known statements over the complex numbers).

The proof relies on a positive characteristic generic vanishing theory developed by Hacon-Patakfalvi, together with a Witt vector version of the Grauert-Riemenschneider vanishing theorem.

 

 

 

 

 

 

 

 

 

 

This talk is part of the activity of the MIUR Excellence Department Projects MathMod@TOV, and the PRIN 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures