Quasi-positive mixed curvature, vanishing theorems, and rational connectedness
Kai Tang
TL;DR
This work introduces the mixed curvature $\mathcal{C}_{a,b}$, a convex combination of the Ricci and holomorphic sectional curvatures, and proves that compact Kahler manifolds with quasi-positive $\mathcal{C}_{a,b}$ are projective when $a>0$ and $3a+2b\ge0$, with $h^{2,0}=0$; if $a,b\ge0$, such manifolds are rationally connected. The authors combine Zhang–Zhang's Bochner-type integral approach with Heier–Wong's uniruledness/rational connectedness framework and employ a Berger averaging technique to derive key integral inequalities. They further derive corollaries for $k$-Ricci curvature, show that quasi-positive $S_2$ implies projectivity (and bounds on the rational dimension), and extend these results to Hermitian manifolds, including vanishing of $H^{2,0}_{\bar\partial}$ in dimension 3 under quasi-positive real bisectional curvature and projectivity under Kahlerity. Overall, the paper extends Yau-type conjectures to the setting of mixed curvature and provides Hermitian-generalized projectivity and rational connectedness criteria, highlighting the robustness of the Bochner-method combined with curvature averaging.
Abstract
In this paper, we consider {\em mixed curvature} $\mathcal{C}_{a,b}$, which is a convex combination of Ricci curvature and holomorphic sectional curvature introduced by Chu-Lee-Tam. We prove that if a compact complex manifold $M$ admits a Kähler metric with quasi-positive mixed curvature and $3a+2b\geq0$, then it is projective. If $a,b\geq0$, then $M$ is rationally connected. As a corollary, the same result holds for $k$-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 have vanishing Hodge number $h^{2,0}$. Furthermore, if it is Kählerian, then it is projective.