Compact Kähler manifolds with nonpositive bisectional curvature

2011/12/07 by Gang Liu, Liu, Gang
Mathematics · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1112.1479

openalex publication_date 2011/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (Mⁿ, g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold N^k with c₁ < 0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.

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