Quasi-positive curvature and projectivity

TL;DR

This paper establishes conditions under which compact Kähler manifolds with certain quasi-positive or non-negative curvature properties are projective and rationally connected, linking curvature conditions to algebraic and topological properties.

Contribution

It proves that specific quasi-positive curvature conditions imply projectivity, and that restricted holonomy combined with non-negative curvature ensures projectivity and rational connectedness.

Findings

Quasi-positive curvature implies projectivity of Kähler manifolds.

Restricted holonomy with non-negative curvature implies projectivity and rational connectedness.

Certain curvature conditions guarantee algebraic and topological properties.

Abstract

In this paper, we first prove that a compact K\"ahler 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 K\"ahler 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 $\mbox{Ric}_k$.