In this article, we prove that for any compact Kähler manifold Mn with real analytic metric and nonpositive bisectional curvature, there exists a finite cover M′ of M such that M′ is a holomorphic and metric fiber bundle over a compact Kähler manifold N with nonpositive bisectional curvature and c1(N) < 0, and the fiber is a flat complex torus. This partially confirms a conjecture of Yau.
"Compact Kähler Manifolds with Nonpositive Bisectional Curvature." J. Differential Geom. 61 (2) 263 - 287, June, 2002. https://doi.org/10.4310/jdg/1090351386