Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.774749
Title: Ricci-flat deformations of orbifolds and asymptotically locally Euclidean manifolds
Author: Lund, Christian Overgaard
ISNI:       0000 0004 7961 9526
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2019
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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 Γ. 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 Γ-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 X̅ and use the deformation theory by Kawamata for the pair (X̅,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 (X̅,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.
Supervisor: Kovalev, Alexei Sponsor: EPSRC ; Cambridge Trust ; DPMMS
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.774749  DOI:
Keywords: moduli space ; Ricci-flat deformations ; Calabi-Yau ; orbifolds ; asymptotically locally Euclidean
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