14
$\begingroup$

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 $i$.

$\endgroup$

1 Answer 1

5
$\begingroup$

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 $A\longrightarrow X$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.