RC-positivity, vanishing theorems and rigidity of holomorphic maps

Abstract

Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle ${\mathcal{O}}_{T_M^{\ast }}(1)$ is not pseudo-effective and ${\mathcal{O}}_{T_N^{\ast }}(1)$ 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.

References in corpus (1)

• A partial converse to the Andreotti-Grauert theorem

Cited by in corpus (1)

• RC-positive metrics on rationally connected manifolds