Kähler manifolds and the curvature operator of the second kind

2022/08/30 by Xiaolong Li, Li, Xiaolong · 3 citations
Mathematics · #53C21 #53C55 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2208.14505

openalex publication_date 2022/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This article aims to investigate the curvature operator of the second kind on Kähler manifolds. The first result states that an m-dimensional Kähler manifold with (3)/(2)(m²-1)-nonnegative (respectively, (3)/(2)(m²-1)-nonpositive) curvature operator of the second kind must have constant nonnegative (respectively, nonpositive) holomorphic sectional curvature. The second result asserts that a closed m-dimensional Kähler manifold with ((3m³-m+2)/(2m))-positive curvature operator of the second kind has positive orthogonal bisectional curvature, thus being biholomorphic to \mathbbCP^m. We also prove that ((3m³+2m²-3m-2)/(2m))-positive curvature operator of the second kind implies positive orthogonal Ricci curvature. Our approach is pointwise and algebraic.

Cited by

• New rigidity theorem of Einstein manifolds and curvature operator of the second kind
• Weitzenböck-Bochner-Kodaira formulas with quadratic curvature terms
• Einstein manifolds under cone conditions for the curvature operator of the second kind

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