Compact K\"ahler manifolds with nonpositive bisectional curvature

TL;DR

This paper proves that compact K"ahler manifolds with nonpositive bisectional curvature have a finite cover structure resembling a flat torus bundle over a negatively curved base, confirming Yau's conjecture and linking curvature to Kodaira dimension.

Contribution

It establishes a biholomorphic and isometric classification of such manifolds, confirming a conjecture of Yau and relating curvature properties to algebraic invariants.

Findings

Finite cover is biholomorphic and isometric to a flat torus bundle over a negatively curved K"ahler manifold.

Kodaira dimension equals the maximal rank of the Ricci tensor for these manifolds.

Global splitting results under certain immersed complex submanifold conditions.

Abstract

Let $(M^n, g)$ be a compact K\"ahler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact K\"ahler manifold $N^k$ with $c_1 < 0$. This confirms a conjecture of Yau. As a corollary, for any compact K\"ahler 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.