Ricci-flat deformations of orbifolds and asymptotically locally Euclidean manifolds

Abstract

In this thesis we study Ricci-flat deformations of Ricci-flat Kähler metrics on compact orbifolds and asymptotically locally Euclidean(ALE) manifolds. In both cases we also study the moduli space of Ricci-flat structures. For this purpose, it is convenient to assume that the initial Ricci-flat metrics are Kähler. Our work extends results by Koiso about Einstein-deformations of Kähler-Einstein metrics on compact manifolds. Orbifolds differ from manifolds by being locally modelled on a quotient of Euclidean space by the action of a finite group $\Gamma$. We adapt a slice construction by Ebin and the Calabi conjecture to orbifolds and show that for compact complex orbifolds with vanishing orbifold first Chern class and all infinitesimal complex deformations integrable, Ricci-flat deformations of Ricci-flat Kähler metric are Kähler, possibly with respect to a perturbed complex structure. We also show that the moduli space of Ricci-flat structures is, up to the action of a finite group, a finite dimensional manifold and we express its dimension in terms of the dimension of certain Dolbeault and sheaf cohomology groups. The strategy is to lift the problem locally to a $\Gamma$-invariant problem on a manifold. ALE manifolds are non-compact manifolds with one end, for which the metric at infinity approximates a flat metric. We study ALE Ricci-flat Kähler manifolds that arise as the complement of a divisor $D$ in a compact Kähler manifold $\bar{X}$ and use the deformation theory by Kawamata for the pair $(\bar{X},D)$. By working with suitably chosen weighted Sobolev and Hölder spaces we recover the relevant elliptic theory for the linearisation of the Ricci operator and the linearisation of the complex Monge-Ampère equation. We prove that integrability of infinitesimal deformations of the pair $(\bar{X},D)$ implies that ALE Ricci-flat deformations of ALE Ricci-flat Kähler metrics are Kähler, possibly with respect to a perturbed complex structure. We also show that the moduli space of ALE Ricci-flat structures is, up to the action of a finite group, a finite dimensional manifold and we express its dimension in terms of the dimension of certain Dolbeault and sheaf cohomology groups.