Reduction of manifolds with semi-negative holomorphic sectional curvature

TL;DR

This paper introduces a new invariant measuring flat directions in the tangent spaces of projective Kähler manifolds with semi-negative holomorphic sectional curvature, relating it to nef and Kodaira dimensions and establishing a splitting theorem.

Contribution

It defines a novel differential geometric numerical invariant for such manifolds and relates it to existing geometric invariants, providing new splitting results.

Findings

The new invariant is bounded above by the nef dimension.

The invariant is bounded below by the numerical Kodaira dimension.

A splitting theorem is established based on the invariant and nef dimension.

Abstract

In this note, we continue the investigation of a projective K\"ahler manifold $M$ of semi-negative holomorphic sectional curvature $H$. We introduce a new differential geometric numerical rank invariant which measures the number of linearly independent {\it truly flat} directions of $H$ in the tangent spaces. We prove that this invariant is bounded above by the nef dimension and bounded below by the numerical Kodaira dimension of $M$. We also prove a splitting theorem for $M$ in terms of the nef dimension and, under some additional hypotheses, in terms of the new rank invariant.