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

Gang Liu*

*Corresponding author for this work

Research output: Contribution to journalArticle

11 Scopus citations

Abstract

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 Kähler manifold with nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions is finitely generated. During the course of the proof, we prove if Mn is a complete noncompact Kähler 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 Kähler manifolds with nonnegative bisectional curvature.

Original languageEnglish (US)
Pages (from-to)775-815
Number of pages41
JournalAnnals of Mathematics
Volume184
Issue number3
DOIs
StatePublished - Jan 1 2016

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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