Quasi-positive mixed curvature, vanishing theorems, and rational connectedness

TL;DR

This paper investigates the implications of quasi-positive mixed curvature on the geometry of compact complex manifolds, establishing conditions under which they are projective or rationally connected, and extending results to Hermitian manifolds.

Contribution

It introduces new vanishing theorems and projectivity criteria based on quasi-positive mixed curvature and generalizes these to Hermitian manifolds.

Findings

Compact manifolds with quasi-positive mixed curvature are projective if 3a+2b≥0.

Such manifolds are rationally connected when a,b≥0.

Quasi-positive 2-scalar curvature implies projectivity.

Abstract

In this paper, we consider {\em mixed curvature} $\mathcal{C}_{a,b}$, which is a convex combination of Ricci curvature and holomorphic sectional curvature introduced by Chu-Lee-Tam. We prove that if a compact complex manifold $M$ admits a K\"{a}hler metric with quasi-positive mixed curvature and $3a+2b\geq0$, then it is projective. If $a,b\geq0$, then $M$ is rationally connected. As a corollary, the same result holds for $k$-Ricci curvature. We also show that any compact K\"{a}hler manifold with quasi-positive 2-scalar curvature is projective. Lastly, we generalize the result to the Hermitian case. In particular, any compact Hermitian threefold with quasi-positive real bisectional curvature have vanishing Hodge number $h^{2,0}$. Furthermore, if it is K\"{a}hlerian, then it is projective.