[Paper Review] On the characterization of abelian varieties in characteristic $p>0$

Christopher D. Hacon, Zsolt Patakfalvi|arXiv (Cornell University)|Feb 4, 2016

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper establishes conditions under which a normal, projective variety $X$ over an algebraically closed field of characteristic $p > 0$ is birational to an abelian variety, by analyzing the Albanese morphism and Kodaira dimension. Using techniques from F-singularities and Cartier modules, it proves that if $\kappa(K_X) = 0$ and the Albanese map is generically finite with degree not divisible by $p$, or if the map is separable and the target abelian variety is ordinary or supersingular, then $X$ is birational to an abelian variety.

ABSTRACT

This article is superseded by 1703.06631. We keep this version here since some of the arguments for the special cases treated here are different than those of 1703.06631.

Motivation & Objective

• To generalize Kawamata-Viehweg's characteristic zero result on varieties of maximal Albanese dimension to positive characteristic.
• To characterize when a normal projective variety $X$ with $\kappa(K_X) = 0$ is birational to an abelian variety in characteristic $p > 0$.
• To extend results from characteristic zero to positive characteristic by adapting tools from F-singularities and generic vanishing theory.
• To treat cases where the Albanese morphism is inseparable, particularly height-1 isogenies, and to handle supersingular and ordinary abelian varieties.
• To define and use the notion of $\kappa(K_X) = 0$ for Weil divisors in the absence of resolution of singularities in positive characteristic.

Proposed method

• Use of the Albanese morphism $a: X \to A$ to reduce the problem to studying the geometry of $X$ over an abelian variety $A$.
• Application of the subadditivity of Kodaira dimension in the context of fibrations to analyze the generic fiber of the Stein factorization.
• Employment of Cartier modules and the $S_0$-construction to analyze the pushforward of the canonical sheaf $a_*\omega_X$.
• Utilization of the $V^0(K_X)$-locus of line bundles with non-vanishing sections to constrain the support of the canonical divisor.
• Leveraging the structure of supersingular and ordinary abelian varieties via Poincaré reducibility and isogenies to elliptic curves.
• Use of the relative Frobenius and foliation theory to analyze the case of finite, purely inseparable Albanese maps of height 1.

Experimental results

Research questions

• RQ1 Under what conditions is a normal projective variety $X$ with $\kappa(K_X) = 0$ over a field of characteristic $p > 0$ birational to an abelian variety?
• RQ2 Can the characteristic zero result on varieties of maximal Albanese dimension be extended to positive characteristic when the Albanese map is not necessarily separable?
• RQ3 How does the behavior of the canonical sheaf and its pushforward under the Albanese morphism constrain the geometry of $X$ in positive characteristic?
• RQ4 What role does the height of the Albanese morphism play in determining birationality to an abelian variety?
• RQ5 Can the structure of the Albanese variety (e.g., ordinary or supersingular) be used to recover birationality results in the absence of separability or degree restrictions?

Key findings

• If the Albanese morphism $a: X \to A$ is generically finite and $p \nmid \deg a$, and $\kappa(K_X) = 0$, then $a$ is birational, so $X$ is birational to an abelian variety.
• If $a$ is separable and $A$ is either ordinary or supersingular, and $\kappa(K_X) = 0$, then $a$ is birational, so $X$ is birational to an abelian variety.
• If $a$ is finite, purely inseparable of height 1, and $X$ is smooth, then $a$ is birational, so $X$ is birational to an abelian variety.
• The proof of Theorem 0.2 shows that if $a$ is separable and $\kappa(K_X) = 0$, then $a$ is surjective, which is a key step toward birationality.
• In the case of supersingular $A$, the Albanese morphism is shown to be étale over a big open subset, implying birationality via purity of the branch locus.
• The canonical sheaf $\omega_X$ is shown to satisfy $c_1(a_*\omega_X) = 0$, which, combined with separability, implies that $a$ is étale over a big open set, leading to birationality.