Varieties with nef anticanonical divisors and Albanese morphisms of relative dimension one in positive characteristic

TL;DR

This work analyzes smooth projective varieties $X$ over an algebraically closed field of characteristic $p>0$ with nef $-K_X$ when the Albanese morphism $a_X$ has relative dimension one. It proves that $a_X$ is necessarily a fibration and provides a precise global structure: in the separable case, after a suitable isogeny base change, $X$ is a projective bundle $\mathbb{P}_A(\mathcal{E})$ with $\mathcal{E}$ a numerically flat rank-2 bundle; in the inseparable case, $X$ arises as a quotient of $A'\times\mathbb{P}^1$ by a rank-one foliation, with a complete description of the foliation and, in dimension three, a full classification. The paper also establishes supporting results on foliations, adjunction, and Frobenius-trace techniques, plus consequences such as non-vanishing of twisted anticanonical sections and semi-ampleness criteria, thereby extending the understanding of nef-anticanonical geometry and Albanese fibrations in positive characteristic. The findings illuminate how inseparability and wild phenomena in characteristic $p$ constrain the global geometry of nef $-K_X$ varieties and yield explicit, canonical models up to isogeny or quotient by foliations, with potential implications for $p$-adic and arithmetic geometry as well as moduli problems.

Abstract

Let $X$ be a smooth projective variety with a nef anticanonical divisor over an algebraically closed field of characteristic $p>0$. In this paper, we establish a precise structure of $X$ under the condition that $a_X: X \to {\rm Alb}(X)$ is of relative dimension one.

Theorems & Definitions (32)

• Theorem 1.1: Theorem \ref{['thm: structure-sep1']} & Theorem \ref{['thm: structure-sep2']}
• Theorem 1.2: Theorem \ref{['thm: structure-inseparable']} & Theorem \ref{['thm: classfy F']}
• Proposition 2.1
• Proposition 2.2
• Theorem 2.3
• proof
• Definition 2.4
• Theorem 2.5: Eji23
• Theorem 2.6
• proof