  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.
