Smooth proper weakly ordinary varieties of maximal Albanese dimension satisfy chi(X, omega_X) >= 0, with chi = 0 for non-general-type examples and the Albanese image then fibered by ordinary abelian varieties.
Generic vanishing theory in positive characteristic
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abstract
We simplify and improve the main fundamental theorems of positive characteristic generic vanishing theory. As a quick corollary of the theory, we prove that a normal variety $X$ of maximal Albanese dimension satisfies $H^0(X, \omega_X) \neq 0$ and that if $\mathrm{Alb}(X)$ is ordinary, then $S^0(X, \omega_X) \neq 0$.
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On the Euler characteristic of weakly ordinary varieties of maximal Albanese dimension
Smooth proper weakly ordinary varieties of maximal Albanese dimension satisfy chi(X, omega_X) >= 0, with chi = 0 for non-general-type examples and the Albanese image then fibered by ordinary abelian varieties.