Gromov-Hausdorff limits of Kähler manifolds and the finite generation conjecture

We study the uniformization conjecture of Yau by using the GromovHausdorff convergence. As a consequence, we confirm Yau’s finite generation conjecture. More precisely, on a complete noncompact Kahler manifold with nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions is finitely generated. During the course of the proof, we prove if M is a complete noncompact Kahler manifold with nonnegative bisectional curvature and maximal volume growth, then M is biholomorphic to an affine algebraic variety. We also confirm a conjecture of Ni on the existence of polynomial growth holomorphic functions on Kahler manifolds with nonnegative bisectional curvature.