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

We show that a smooth proper weakly ordinary variety $X$ of maximal Albanese dimension satisfies $\chi (X,{\omega }_X)\ge 0$. 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.