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Einstein deformations of Kähler Einstein metrics

Paul-Andi Nagy

Abstract

We study Einstein deformations of negative Kähler Einstein metrics. We relate the second order Einstein deformation theory of negative Kähler-Einstein metrics to the complex geometry of the underlying Kähler manifold. After suitable gauge normalisation we show that the Taylor expansion to order two of an Einstein deformation tangent to $h_1$ in the infinitesimal deformation space is fully determined by $h_1^2$ and the divergence of the Kodaira-Spencer bracket $[h_1,h_1]^c$. This substantially refines and extends recent results of Nagy-Semmelmann which state that Einstein deformations for negative Kähler-Einstein metrics are unobstructed to second order.

Einstein deformations of Kähler Einstein metrics

Abstract

We study Einstein deformations of negative Kähler Einstein metrics. We relate the second order Einstein deformation theory of negative Kähler-Einstein metrics to the complex geometry of the underlying Kähler manifold. After suitable gauge normalisation we show that the Taylor expansion to order two of an Einstein deformation tangent to in the infinitesimal deformation space is fully determined by and the divergence of the Kodaira-Spencer bracket . This substantially refines and extends recent results of Nagy-Semmelmann which state that Einstein deformations for negative Kähler-Einstein metrics are unobstructed to second order.
Paper Structure (14 sections, 32 theorems, 144 equations)

This paper contains 14 sections, 32 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Key arguments and outline of proofs for Kähler metrics
  5. Preliminaries
  6. The Frölicher-Nijenhuis bracket
  7. Bianchi map normalisation to second order
  8. Role of the trace
  9. Proof of normalisation to second order
  10. Negative Kähler-Einstein metrics
  11. Elements of Kähler geometry
  12. Type decomposition of $[\cdot,\cdot]^{\mathrm{FN}}$ and the Kodaira-Spencer bracket
  13. Structure of the operator $\mathbf{v}$
  14. Proof of Theorem \ref{['main-1i']}

Key Result

Theorem 1.1

NS-E[Theorems 3.13 and 4.3] The Einstein equation to second order is given by In particular the obstruction to second order deformation reads

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 57 more