The geometry of complete positively curved Kahler manifolds

This thesis is concerned with a conjecture raised by Yau Shing-Tung in early 1970's in complex differential geometry: any complete Kähler manifold of positive holomorphic bisectional curvature is biholomorphic to the complex Euclidean space Cn. The proof of this conjecture relies on the understanding of the geometry of the manifold. We study the curvature decay and the volume growth of these manifolds as well as the relationship between them. Basically, there should be a one to one correspondence of these two things. We prove that as the growth of the volume varies from Euclidean to half Euclidean, the curvature decays from quadratically to linearly. Based on this knowledge, we construct holomorphic functions or holomorphic sections of holomorphic line bundles of suitable growth, so that the manifold can be embedded as an algebraic subvariety. The principle difficulty arises from controlling the dimension of the holomorphic objects constructed, and this reduces to some multiplicity estimate and eventually depends the geometry of the underlying manifold. The main results we prove are the following: if M is a complete Kähler manifold with positive bounded sectional curvature with finite first Chern-number, then M is biholomorphic to a quasi-projective algebraic variety; if M has maximal volume growth and nonnegative curvature (nonnegative bisectional curvature for complex dimension 2, nonnegative curvature operator for 3 up), then M is biholomorphic to a quasi-affine algebraic variety. Both results imply that under above assumption for n = 2, M is biholomorphic to C2.