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Curvature Operators on Kähler Manifolds

Barry Minemyer

Abstract

We prove that there exist Kähler manifolds that are not homotopy equivalent to a quotient of complex hyperbolic space but which admit a Riemannian metric with nonpositive curvature operator. This shows that Kähler manifolds do not satisfy the same type of rigidity with respect to the curvature operator as quaternionic hyperbolic and Cayley hyperbolic manifolds and are thus more similar to real hyperbolic manifolds in this setting. Along the way we also calculate explicit values for the eigenvalues of the curvature operator with respect to the standard complex hyperbolic metric.

Curvature Operators on Kähler Manifolds

Abstract

We prove that there exist Kähler manifolds that are not homotopy equivalent to a quotient of complex hyperbolic space but which admit a Riemannian metric with nonpositive curvature operator. This shows that Kähler manifolds do not satisfy the same type of rigidity with respect to the curvature operator as quaternionic hyperbolic and Cayley hyperbolic manifolds and are thus more similar to real hyperbolic manifolds in this setting. Along the way we also calculate explicit values for the eigenvalues of the curvature operator with respect to the standard complex hyperbolic metric.
Paper Structure (8 sections, 6 theorems, 35 equations)

This paper contains 8 sections, 6 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Curvature formulas for the complex hyperbolic metric written in polar coordinates about a complex hyperplane
  3. The metric on $\mathbb{C} \mathbb{H}^n$ in polar coordinates about $\mathbb{C} \mathbb{H}^{n-1}$
  4. Eigenvalues for the curvature operator associated to $\mu$
  5. The n=2 case
  6. The n=3 case
  7. The case for general $n$
  8. Overview of the metric and proof of the main theorem

Key Result

Theorem 1.1

For each complex dimension $n$ there exist Kähler manifolds $M$ of dimension $n$ which admit a Riemannian metric with nonpositive curvature operator and are not homotopy equivalent to a quotient of $\mathbb{C} \mathbb{H}^n$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1: compare Sections 7 and 8 of Bel complex, Theorem 4.3 of Min warped, and Theorem 2.2 of Min CH
  • Lemma 2.2: See Section 2 of Min CH
  • Theorem 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:main theorem']}
  • proof : Proof of Theorem \ref{['thm:main cor']}