Published online by Cambridge University Press: 11 October 2019
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.
This work was partially supported by China’s Recruitment Program of Global Experts and NSFC 11688101.