Einstein deformations of K\"ahler Einstein metrics

TL;DR

This paper investigates the second order Einstein deformation theory of negative Kähler Einstein metrics, revealing how the deformation's Taylor expansion is determined by specific geometric quantities, thus extending previous unobstructedness results.

Contribution

It establishes a detailed relationship between second order Einstein deformations and the complex geometry of the underlying Kähler manifold, refining prior unobstructedness results.

Findings

Second order deformation expansion is determined by the square of the initial deformation and the divergence of the Kodaira-Spencer bracket.

The results extend recent findings by Nagy-Semmelmann on unobstructed Einstein deformations.

Provides a deeper understanding of the deformation space structure for negative Kähler Einstein metrics.

Abstract

We study Einstein deformations of negative K\"ahler Einstein metrics. We relate the second order Einstein deformation theory of negative K\"ahler-Einstein metrics to the complex geometry of the underlying K\"ahler 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 K\"ahler-Einstein metrics are unobstructed to second order.