Characterizations of Abelian varieties (3-folds) in positive characteristic

Question

From this question Characterizations of complex Abelian varieties (especially 3-folds) among projective nonsingular varieties? I learned that if $X$ is a smooth complex projective variety of dimension $g$, then $X$ is a torsor over an Abelian variety (its Albanese variety) if and only if $\omega_X \cong \mathcal{O}_X$ and ${\rm h^1}(X; \mathcal{O}_X) = g$.

I would like to know the corresponding statement over an algebraically closed field of positive characteristic. If necessary, we could exclude small primes:

By the Bombieri-Mumford classification of surfaces in positive characteristic, I know that the above statement holds for $g=2$ and characteristics different from 2 and 3. In those small characteristics, one also gets examples of "quasi-hyperelliptic surfaces" (essentially because the Albanese could be a non-reduced group scheme), and they distinguish the two classes via étale cohomology (which as far as I can tell, I cannot calculate for my examples of interest).

For my own purposes, I want the answer for $g=3$, but the answer in general is also welcome. If it helps, I also know that ${\rm h}^i(X; \mathcal{O}_X) = \binom{g}{i}$ for all $i$.

Answer 1

I don't have an answer, but a couple of comments that may be useful:

The group $H^1({\cal O}_X)$ can be identified with the Zariski tangent space of ${\rm Pic}^0(X)$. Since the latter can be non-reduced in characteristic $p$ (in characteristic zero, this is impossible by a theorem of Cartier), the dimension of $H^1({\cal O}_X)$ may be larger than $\dim{\rm Alb}(X)=\dim {\rm Pic}^0(X)=b_1(X)/2$. So maybe you would want to ask whether $$ \omega_X\cong{\cal O}_X, b_1(X)=2g $$ implies that $X$ is a torsor over an Abelian variety. For example, quasi-hyperelliptic surfaces with non-reduced ${\rm Pic}^0(X)$ satisfy $\omega_X\cong{\cal O}_X$, $h^1({\cal O}_X)=2$, but their Albanese variety is $1$-dimensional.

Let $f:X\to{\rm Alb}(X)$ be the Albanese morphism. By a result of Igusa, the pull-back map of global $1$-forms $$ f^*:H^0(\Omega^1_{\rm Alb X}) \rightarrow H^0(\Omega_X^1) $$ has no kernel, which might be useful. ${\bf Correction:}$ However, even if $f$ is generically finite, Igusa's result does not imply that $f$ is generically etale. In fact, in every positive characteristic $p$, there do exist surfaces of general type, whose Albanese morphisms are purely inseparable of degree $p$ onto an Abelian surface (details upon request!). In these examples, the kernel of $f^*\Omega_{\rm Alb X}^1\to\Omega_X^1$ contains a rank $1$ subsheaf.

Varieties with trivial tangent bundles in characteristic $p$ were studied by Mehta, Nori, and Srinivas (Compositio Math. 64 (1987), no. 2, 191-212), and in case these varieties are moreover assumed to be ordinary (which one should think of as the "generic" or "nice" case), then there exists an Abelian variety $A$ and an etale Galois cover $A\longrightarrow X$.