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

Gang Liu

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arXiv · 1501.00681

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

Gang Liu

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Abstract

We study the uniformization conjecture of Yau by using the Gromov-Haudorff 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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BibTeX RIS

Gang Liu. 2015-05-02. Gromov-Hausdorff limit of Kähler manifolds and the finite generation conjecture. https://arxiv.org/abs/1501.00681

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