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Algebraic fibre spaces with strictly nef relative anti-log canonical divisor

Jie Liu, Wenhao Ou, Juanyong Wang, Xiaokui Yang, Guolei Zhong

Abstract

Let $(X,Δ)$ be a projective klt pair, and $f:X\to Y$ a fibration to a smooth projective variety $Y$ with strictly nef relative anti-log canonical divisor $-(K_{X/Y}+Δ)$. We prove that $f$ is a locally constant fibration with rationally connected fibres, and the base $Y$ is a canonically polarized hyperbolic projective manifold. In particular, when $Y$ is a single point, we establish that $X$ is rationally connected. Moreover, when $\dim X=3$ and $-(K_X+Δ)$ is strictly nef, we prove that $-(K_X+Δ)$ is ample, which confirms the singular version of a conjecture of Campana-Peternell for threefolds.

Algebraic fibre spaces with strictly nef relative anti-log canonical divisor

Abstract

Let be a projective klt pair, and a fibration to a smooth projective variety with strictly nef relative anti-log canonical divisor . We prove that is a locally constant fibration with rationally connected fibres, and the base is a canonically polarized hyperbolic projective manifold. In particular, when is a single point, we establish that is rationally connected. Moreover, when and is strictly nef, we prove that is ample, which confirms the singular version of a conjecture of Campana-Peternell for threefolds.

Paper Structure

This paper contains 8 sections, 38 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Preliminary results
  4. Periodic points of a special Kähler group
  5. Proofs of text, text and text
  6. Ampleness of text, Proof of text
  7. text-effectivity of text
  8. Proof of text

Key Result

Theorem 1

Let $X$ be a normal projective variety. Suppose that there is an effective $\mathbb Q$-divisor $\varDelta$ on $X$ such that the pair $(X,\varDelta)$ is klt and that $-(K_X+\varDelta)$ is strictly nef. Then $X$ is rationally connected. In particular, the augmented irregularity $q^\circ(X)=0$.

Theorems & Definitions (81)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1
  • Theorem 1.3: Araujo-Druel
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Definition 2.1: cf. CCP08
  • Proposition 2.2
  • proof
  • ...and 71 more