{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/291967"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/291967","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"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.","abstract_html":"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.","abstract_has_math":true,"creators":["Lund, Christian Overgaard"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Kovalev, Alexei"],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-07-19","date_published":"2019-07-19","updated_at":"2026-07-22T22:24:04Z","subjects":["moduli space","Ricci-flat deformations","Calabi-Yau","orbifolds","asymptotically locally Euclidean"],"languages":["en"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/4ec14916-0ca1-402d-8782-107e750c7ec7/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.39121","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Kovalev, Alexei"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["EPSRC. 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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."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["cbb546d5f9d7000ed07bb6d05c8f7c2b","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["Ricci-flat deformations of orbifolds and asymptotically locally Euclidean manifolds"]}]}],"canonical_facts":{"dc:contributor.advisor":["Kovalev, Alexei"],"dc:contributor.sponsor":["EPSRC. Cambridge Trust. DPMMS"],"dc:creator":["Lund, Christian Overgaard"],"dc:date.issued":["2019-07-19"],"dc:description.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. 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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. 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