Rational curves on Hermitian manifolds

TL;DR

This paper proves the existence of rational curves on compact Hermitian manifolds with positive holomorphic bisectional curvature, confirming a question posed by S.-T. Yau, and discusses exceptions like the Hopf surface.

Contribution

It establishes the presence of rational curves on certain Hermitian manifolds with positive curvature, extending Mori's results to a broader class.

Findings

Rational curves exist on compact Hermitian manifolds with positive holomorphic bisectional curvature.

The Hopf surface with non-negative curvature contains no rational curve.

Confirms a question of S.-T. Yau regarding rational curves on Hermitian manifolds.

Abstract

By using analytic method, we prove that there exist rational curves on compact Hermitian manifolds with positive holomorphic bisectional curvature. It confirms a question of S.-T. Yau. It is well-known that Mori proved in \cite{Mori79} that every compact complex manifold $N$ with $c_1(N)>0$ contains at least one rational curve. However, as a borderline example, we show that the standard Hopf surface $S^1\times S^3$ has a Hermitian metric with non-negative holomorphic bisectional curvature (in particular, $c_1(S^1\times S^3)\geq 0$), but it contains no rational curve.