Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
- $Y$ is an abelian variety, or
- the étale fundamental group of $Y$ is trivial?
Motivation:
The Beauville-Bogomolov decomposition theorem for $\mathbb C$ dictates that a smooth projective variety $X/\mathbb C$ with $K_X$ trivial has a finite étale cover $Y \rightarrow X$ such that $Y$ is a product of abelian varieties, "strict" Calabi–Yau varieties and irreducible symplectic varieties. The two latter has their étale fundamental groups trivial.
Though there are no universally agreed definition of irreducible symplectic varieties in char. p, we could still "disprove" Beauville-Bogomolov decomposition in char. p if the covering spaces have undesirable properties.
Best Answer
From pp. 488-489 of W.E. Lang's thesis (Ann. ENS. vol. 12, 1979) it follows that neither case can arise. For the fibration $X\to C$ has an elliptic base and all its geometric fibers are cuspidal rational curves, so that $\pi_1(X) =\pi_1(C)$ and then every etale cover of $X$ is also a quasihyperelliptic surface.