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

• 成果类型: Article
• 署名作者: Liu, Gang
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2016.184.3.4
• 发表日期: 2016
• 页码: 775-815
• 关键词: poincare-lelong equation; RICCI CURVATURE; SPACES; uniformization; SURFACES; THEOREM

摘要

We study the uniformization conjecture of Yau by using the Gromov-Hausdorff 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-n 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.