Third order Einstein deformations for Kaehler-Einstein metrics

Abstract

For compact Kähler manifolds $(M,g,J)$ with negative scalar curvature we study the existence problem for non-trivial Einstein deformations of $g$, that is small time curves $g_t$ of Einstein metrics with $g_0=g$. No asssumption on the complex structure $J$ is made; also we do not assume that the metrics $g_t$ are Kähler w.r.t. $J$. We determine explicitly the obstruction to third order Einstein deformation for $g$; that is we fully solve the equations $(\Ric^{g_t})^{(k)}(0)=0$ for $1 \leq k \le 3$ in terms of the Taylor expansion $g^{-1}g_t=\id+th_1+\tfrac{t^2}{2!}h_2+\tfrac{t^3}{3!}h_3+o(t^4)$ at $t=0$. Up to a suitable gauge transformation we show that third order integrability for the Einstein equation amounts to Maurer-Cartan type equations and polynomial identities relating the coefficients $h_3,h_2,h_1$. This result is interpreted in terms of the underlying complex geometry of $M$ by means of the Cayley transform of the metric $g$; the Cayley transform is also used for formulating conjectures for the higher order Einstein deformation problem.