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Complete Kähler manifolds with nonnegative Ricci curvature II

Gang Liu

TL;DR

The paper addresses rigidity questions for complete noncompact Kähler manifolds with nonnegative Ricci curvature under Euclidean volume growth and quadratic curvature decay. It analyzes tangent cones at infinity, proving that the $Q$-Gorenstein property is preserved across all cones and that the holomorphic spectrum is constant in complex dimension two. Key technical contributions include constructing homogeneous canonical forms on tangent cones via pullbacks and applying Hörmander $L^2$ estimates to extend holomorphic sections, leading to finite limits of curvature-integral expressions. Consequently, in complex dimension two, the manifold is biholomorphic to the resolution of an affine algebraic variety, bridging differential geometry and algebraic geometry.

Abstract

We study rigidity on certain Kähler manifolds with nonnegative Ricci curvature. Among others things, we show that a complete noncompact Kähler surface with nonnegative Ricci curvature, Euclidean volume growth and quadratic curvature decay is biholomorphic to resolution of an affine algebraic variety.

Complete Kähler manifolds with nonnegative Ricci curvature II

TL;DR

The paper addresses rigidity questions for complete noncompact Kähler manifolds with nonnegative Ricci curvature under Euclidean volume growth and quadratic curvature decay. It analyzes tangent cones at infinity, proving that the -Gorenstein property is preserved across all cones and that the holomorphic spectrum is constant in complex dimension two. Key technical contributions include constructing homogeneous canonical forms on tangent cones via pullbacks and applying Hörmander estimates to extend holomorphic sections, leading to finite limits of curvature-integral expressions. Consequently, in complex dimension two, the manifold is biholomorphic to the resolution of an affine algebraic variety, bridging differential geometry and algebraic geometry.

Abstract

We study rigidity on certain Kähler manifolds with nonnegative Ricci curvature. Among others things, we show that a complete noncompact Kähler surface with nonnegative Ricci curvature, Euclidean volume growth and quadratic curvature decay is biholomorphic to resolution of an affine algebraic variety.

Paper Structure

This paper contains 2 sections, 7 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. The proofs

Key Result

Theorem 1

Let $M^n (n\geq 2)$ be a complete noncompact Kähler manifold with nonnegative Ricci curvature, Euclidean volume growth and quadratic curvature decay. Assume one tangent cone at infinity is Q-Gorenstein, then all tangent cones are Q-Gorenstein. Moreover, for all $k>0$, $\lim\limits_{r\to\infty}r^{2k-

Theorems & Definitions (13)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Remark
  • proof
  • Proposition 1
  • Claim 1
  • proof
  • Lemma 1
  • ...and 3 more