K\"ahler manifolds and mixed curvature

TL;DR

This paper investigates the geometric properties of compact K"ahler manifolds with various mixed curvature conditions, establishing links between curvature negativity and the positivity of the canonical line bundle, with implications for projectivity and simply connectedness.

Contribution

It introduces new results connecting mixed curvature bounds to the nefness, bigness, and ampleness of the canonical line bundle, extending previous work to conformally K"ahler manifolds.

Findings

Canonical line bundle is nef under non-positive mixed curvature.

Manifold is projective with big and nef canonical bundle if curvature is negative at some point.

Canonical line bundle is ample if the curvature is negative everywhere.

Abstract

In this work we consider compact K\"ahler manifolds with non-positive mixed curvature which is a "convex combination" of Ricci curvature and holomorphic sectional curvature. We show that in this case, the canonical line bundle is nef. Moreover, if the curvature is negative at some point, then the manifold is projective with canonical line bundle being big and nef. If in addition the curvature is negative, then the canonical line bundle is ample. As an application, we answer a question of Ni concerning manifolds with negative $k$-Ricci curvature and generalize a result of Wu-Yau and Diverio-Trapani to the conformally K\"ahler case. We also show that the compact K\"ahler manifold is projective and simply connected if the mixed curvature is positive.