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

2024/05/06 by Kai Tang, Tang, Kai · 2 citations
Computer Science · Mathematics · #53C55 #Advanced Banach Space Theory #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2405.03895

openalex publication_date 2024/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

In this paper, we consider \em mixed curvature 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≥0, then it is projective. If a,b≥0, 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.

Cited by

• On Kodaira dimension and scalar curvature in almost Hermitian geometry
• Constant kth-mixed curvature

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