Quasi‐positive mixed curvature, vanishing theorems, and rational connectedness

Abstract In this paper, we consider mixed curvature , which is a convex combination of Ricci curvature and holomorphic sectional curvature introduced by Chu–Lee–Tam [Trans. Amer. Math. Soc. 375 (2022), no. 11, 7925‐7944]. We prove that if a compact complex manifold admits a Kähler metric with quasi‐positive mixed curvature and , then it is projective. If , then is rationally connected. As a corollary, the same result holds for ‐Ricci curvature. We also show that any compact Kähler manifold with quasi‐positive 2‐scalar curvature is projective. Lastly, we generalize the result to the Hermitian case. In particular, any compact Hermitian threefold with quasi‐positive real bisectional curvature has vanishing Hodge number . Furthermore, if it is Kählerian, then it is projective.