RC-positivity, vanishing theorems and rigidity of holomorphic maps

TL;DR

This paper establishes conditions under which holomorphic maps between compact complex manifolds must be constant, linking geometric positivity properties of tangent and cotangent bundles to map rigidity.

Contribution

It proves that certain positivity conditions on tangent and cotangent bundles imply the non-existence of non-constant holomorphic maps between compact complex manifolds.

Findings

No non-constant holomorphic maps from manifolds with RC-positive tangent bundles to those with nef cotangent bundles.

Holomorphic maps from manifolds with positive holomorphic sectional curvature to those with non-positive holomorphic bisectional curvature are trivial.

The results connect geometric positivity properties to the rigidity of holomorphic maps.

Abstract

Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle $\mathscr{O}_{T_M^*}(1)$ is not pseudo-effective and $\mathscr{O}_{T_N^*}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.