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

Pages 775-815 from Volume 184 (2016), Issue 3 by Gang Liu

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 $M^n$ 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.

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Keywords
, ,
Mathematical Subject Classification
Primary:
DOI
https://doi.org/10.4007/annals.2016.184.3.4
MR
3549623
zbMATH
06647932
Milestones
Received: 3 January 2015
Revised: 15 October 2015
Accepted: 14 December 2015
Published online: 16 September 2016

Authors

Gang Liu

Northwestern University, Evanston, IL