Birational characterization of Abelian varieties and ordinary Abelian varieties in characteristic p>0

Abstract

Let X/k be an abelian variety over an algebraically closed field k of characteristic p > 0. In this paper, using the Azumaya property of the sheaf of crystalline differential operators and the Morita equivalence, we show that etale locally over the Hitchin base, the moduli stack of Higgs bundles on the Frobenius twist X is equivalent to that of local systems on X. We follow the approach of [Gro16].

Generic vanishing in characteristic p>0 and the geometry of theta divisors

“…Nevertheless recently, using the theory of Cartier modules and Serre vanishing, a technical generalization of generic vanishing was proven in [HP16] for projective varieties over a field of characteristic p > 0. Despite its technical nature, this result leads the way to some remarkable geometric applications such as a characterization of (ordinary) abelian varieties [HPZ19]. In this paper we further refine the results of [HP16] proving a more precise generic vanishing result in characteristic p > 0 and we use this result to investigate the geometry of pluri-theta divisors.…”

Abundance for non‐uniruled 3‐folds with non‐trivial Albanese maps in positive characteristics

“…As an easy application, we can characterize abelian varieties birationally by the conditions. Note that it is expected that a smooth projective variety, with zero Kodaira dimension and maximal Albanese dimension, is birational to an abelian variety, which is finally proved in [17] by much more technical arguments later. Moreover if κ(X) = 0 and either (a) the Albanese map a X : X → A X factors into a X = a X • σ : X X → A X where σ : X X is a birational map to a minimal model X of X with at most klt singularities, and K X ∼ Q 0, or (b) char k = p > 5, dim X = 3 and A X is simple, that is, A X contains no non-trivial abelian varieties [26,IV.…”

“…As an easy application, we can characterize abelian varieties birationally by the conditions. Note that it is expected that a smooth projective variety, with zero Kodaira dimension and maximal Albanese dimension, is birational to an abelian variety, which is finally proved in by much more technical arguments later. Theorem Let be a smooth projective variety of maximal Albanese dimension of dimension .…”

Frobenius techniques in birational geometry

“…Then in [HP13, Thm 1.1.1.a] it is shown that a smooth projective variety X is birational to an ordinary abelian variety if and only if its first Betti number is 2 dim X, κ S (X) = 0 (where κ S is the version of Kodaira dimension defined using S 0 ( ) instead of H 0 ( ) [HP13, 4.1]), and the degree of the Albanese map of X is prime-to-p. Similar methods yielded also a characterization of abelian varieties in [HP15] (so no ordinarity): X is birational to an abelian variety if and only if κ(X) = 0 and the Albanese map is generically finite of degree prime-to-p over its image. Furthermore, using the results of [HP13], Sannai and Tanaka proved another characterization of ordinary ablian varieties [ST16]: X is an ordinary abelian variety if and only if K X is pseudo-effective and F e * O X is a direct sum of invertible sheaves for infinitely many integers e > 0.…”