Ricci-flat deformations of orbifolds and asymptotically locally Euclidean manifolds

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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2019

Author dc:creator

• Lund, Christian Overgaard

Advisor dc:contributor.advisor

• Kovalev, Alexei

• moduli space
• Ricci-flat deformations
• Calabi-Yau
• orbifolds
• asymptotically locally Euclidean

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/4ec14916-0ca1-402d-8782-107e750c7ec7/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

en

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.39121

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/291967