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Quasi-positive curvature and projectivity

Yiyang Du, Yanyan Niu

Abstract

In this paper, we first prove that a compact Kähler manifold is projective if it satisfies certain quasi-positive curvature conditions, including quasi-positive $S_2^\perp,\, S_2^+,\,\mbox{Ric}_3^\perp, \,\mbox{Ric}_3^+$ or $2$-quasi-positive $\mbox{Ric}_k$. Subsequently, we prove that a compact Kähler manifold with a restricted holonomy group is both projective and rationally conected if it satisfies some non-negative curvature condition, including non-negative $S_2^\perp,\, S_2^+,\,\mbox{Ric}_3^\perp, \,\mbox{Ric}_3^+$ or $2$-non-negative $\mbox{Ric}_k$.

Quasi-positive curvature and projectivity

Abstract

In this paper, we first prove that a compact Kähler manifold is projective if it satisfies certain quasi-positive curvature conditions, including quasi-positive or -quasi-positive . Subsequently, we prove that a compact Kähler manifold with a restricted holonomy group is both projective and rationally conected if it satisfies some non-negative curvature condition, including non-negative or -non-negative .

Paper Structure

This paper contains 7 sections, 4 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Curvature conditions
  4. Fundamental identity
  5. Proof of Theorem \ref{['thm1']} and Theorem \ref{['thm2']}
  6. Proof of Theorem \ref{['thm1']}
  7. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1.1

Let $(M^n, g)\,(n\ge 2)$ be a compact Kähler manifold with one of the following curvature conditions, Then $h^{2,0}=0$. In particular, $M$ is projective.

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2: ZZ
  • proof
  • Remark 2.3