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On compact Kähler manifold with strongly pseudo-effective tangent bundle

Xiaojun Wu

Abstract

In the first part of this note, we discuss the compact Kähler manifold with a strongly pseudo-effective tangent bundle. In the second part, we give new proof of the fact that the only projective manifolds with the big tangent bundle are the projective spaces. In the third part, we give a characterisation of nef vector bundle.

On compact Kähler manifold with strongly pseudo-effective tangent bundle

Abstract

In the first part of this note, we discuss the compact Kähler manifold with a strongly pseudo-effective tangent bundle. In the second part, we give new proof of the fact that the only projective manifolds with the big tangent bundle are the projective spaces. In the third part, we give a characterisation of nef vector bundle.

Paper Structure

This paper contains 5 sections, 23 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Fundamental group of manifolds with strongly psef tangent bundle
  3. Characterisation of projective space
  4. Characterisation of nef vector bundle
  5. Appendix. Surjectivity of Albanese morphism

Key Result

Proposition 1

Let $X$ be a projective manifold such that $T_X$ is strongly psef. Let $\tilde{X}$ be a finite étale cover of maximum irregularity $q(\tilde{X} ) = \tilde{q}(X)$. Then the Albanese morphism of $\tilde{X}$ is a smooth fibration such that the very general fibre $F$ has a strongly psef tangent bundle a

Theorems & Definitions (66)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • ...and 56 more