On the Euler characteristic of weakly ordinary varieties of maximal Albanese dimension

TL;DR

This paper proves that weakly ordinary varieties of maximal Albanese dimension have non-negative Euler characteristic, and characterizes those not of general type as having zero Euler characteristic with Albanese images fibered by abelian varieties.

Contribution

It establishes new bounds on the Euler characteristic for a class of varieties using positive characteristic vanishing theorems and Witt vector techniques.

Findings

hi(X, _X)  0 for weakly ordinary varieties of maximal Albanese dimension

If not of general type, then hi(X, _X) = 0 and the Albanese image is fibered by abelian varieties

Uses positive characteristic generic vanishing and Witt vector Grauert-Riemenschneider vanishing techniques

Abstract

We show that a smooth proper weakly ordinary variety $X$ of maximal Albanese dimension satisfies $\chi(X, \omega_X) \geq 0$. We also show that if $X$ is not of general type, then $\chi(X, \omega_X) = 0$ and the Albanese image of $X$ is fibered by abelian varieties. The proof uses the positive characteristic generic vanishing theory developed by Hacon-Patakfalvi, as well as our recent Witt vector version of Grauert-Riemenschneider vanishing.