Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Irregular threefolds with numerically trivial canonical divisor

Jingshan Chen, Chongning Wang, Lei Zhang

TL;DR

The paper advances the understanding of irregular $K$-trivial varieties in positive characteristic by exploiting the Albanese morphism to reveal a transversal bi-fibration structure, allowing explicit descriptions of irregular threefolds. It develops a robust toolkit—Frobenius techniques, foliation push-down/pullback, flat-base-change behavior, and semi-ample divisor constructions—to produce second, Albanese-transversal fibrations and classify cases where the Albanese image has dimension 1 or 2. The authors provide a detailed breakdown into smooth, quasi-elliptic, and inseparable fibrations, delivering explicit classifications for Case (C1), (C2), and (C3) and a comprehensive structure theorem for irregular $K$-trivial threefolds, including concrete examples and foliations. An important byproduct is an effectivity bound for pluricanonical maps in characteristic $p$, notably when the Albanese dimension is 2 or when $q=2$, linking geometric structure to torsion properties of $K_X$. The results extend the geometric picture of $K$-trivial varieties beyond characteristic zero, offering a framework applicable to higher-dimensional irregular $K$-trivials under similar Albanese-dimension constraints.

Abstract

In this paper, we classify irregular threefolds with numerically trivial canonical divisors in positive characteristic. For such a variety, if its Albanese dimension is not maximal, then the Albanese morphism will induce a fibration which either maps to a curve or is fibered by curves. In practice, we treat arbitrary dimensional irregular varieties with either one dimensional Albanese fiber or one dimensional Albanese image. We prove that such a variety carries another fibration transversal to its Albanese morphism (a "bi-fibration" structure), which is an analog structure of bielliptic or quasi-bielliptic surfaces. In turn, we give an explicit description of irregular threefolds with trivial canonical divisors.

Irregular threefolds with numerically trivial canonical divisor

TL;DR

The paper advances the understanding of irregular -trivial varieties in positive characteristic by exploiting the Albanese morphism to reveal a transversal bi-fibration structure, allowing explicit descriptions of irregular threefolds. It develops a robust toolkit—Frobenius techniques, foliation push-down/pullback, flat-base-change behavior, and semi-ample divisor constructions—to produce second, Albanese-transversal fibrations and classify cases where the Albanese image has dimension 1 or 2. The authors provide a detailed breakdown into smooth, quasi-elliptic, and inseparable fibrations, delivering explicit classifications for Case (C1), (C2), and (C3) and a comprehensive structure theorem for irregular -trivial threefolds, including concrete examples and foliations. An important byproduct is an effectivity bound for pluricanonical maps in characteristic , notably when the Albanese dimension is 2 or when , linking geometric structure to torsion properties of . The results extend the geometric picture of -trivial varieties beyond characteristic zero, offering a framework applicable to higher-dimensional irregular -trivials under similar Albanese-dimension constraints.

Abstract

In this paper, we classify irregular threefolds with numerically trivial canonical divisors in positive characteristic. For such a variety, if its Albanese dimension is not maximal, then the Albanese morphism will induce a fibration which either maps to a curve or is fibered by curves. In practice, we treat arbitrary dimensional irregular varieties with either one dimensional Albanese fiber or one dimensional Albanese image. We prove that such a variety carries another fibration transversal to its Albanese morphism (a "bi-fibration" structure), which is an analog structure of bielliptic or quasi-bielliptic surfaces. In turn, we give an explicit description of irregular threefolds with trivial canonical divisors.
Paper Structure (37 sections, 31 theorems, 72 equations)

This paper contains 37 sections, 31 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Effectivity of the pluricanonical map of threefolds
  3. Notation and conventions
  4. Preparations
  5. Frobenius morphisms
  6. Foliations and purely inseparable morphisms
  7. "Pushing down" and pullback foliations
  8. "Pushing-down" foliations along a fibration
  9. Pullback of a foliation
  10. A property of fibred varieties under flat base changes
  11. Behavior of the relative canonical divisor under purely inseparable base changes
  12. Numerical dimension
  13. Covering theorem
  14. Adjunction formula and characterization of abelian varieties
  15. Some results of elliptic fibrations
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ be a normal $\mathbb Q$-factorial projective threefold over $k$ with $K_X\equiv0$. Denote the Albanese morphism of $X$ by $a_X\colon X\to A$, and assume $\dim a_X(X) >0$. Then the following statements hold.

Theorems & Definitions (55)

  • Theorem 1.1: = Theorem \ref{['thm:3fold']}
  • Remark 1.2
  • Corollary 1.3: see Section \ref{['sec:effectivity']}
  • Proposition 2.1: CWZ23
  • Lemma 2.2: Posva24
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 45 more