On the construction of asymptotically conical Calabi-Yau manifolds

This thesis is concerned with the construction of asymptotically conical (AC) Calabi-Yau manifolds. We provide an alternative proof of a result by Goto that states that the basic (p, 0)-Hodge numbers of a positive Sasaki manifold vanish for p > 0. Our main theorem then gives sufficient conditions...

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Main Author: Conlon, Ronan Joseph
Other Authors: Haskins, Mark
Published: Imperial College London 2011
Subjects:
510
Online Access:http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.540652
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spelling ndltd-bl.uk-oai-ethos.bl.uk-5406522017-08-30T03:16:38ZOn the construction of asymptotically conical Calabi-Yau manifoldsConlon, Ronan JosephHaskins, Mark2011This thesis is concerned with the construction of asymptotically conical (AC) Calabi-Yau manifolds. We provide an alternative proof of a result by Goto that states that the basic (p, 0)-Hodge numbers of a positive Sasaki manifold vanish for p > 0. Our main theorem then gives sufficient conditions on a non-compact Kähler manifold to admit an AC Calabi-Yau metric in each compactly supported Kähler class. As a corollary to this, we recover a result of van Coevering which guarantees the existence of an AC Calabi-Yau metric in each compactly supported Kähler class of a crepant resolution of a Calabi-Yau cone. It also follows that we are able to give sufficient conditions on a pair (X, D) , where X is a compact Kähler manifold and D is a divisor supporting the anti-canonical bundle of X, for X\D to admit an AC Calabi-Yau metric in each compactly supported Kähler class. We extend this last result to include cohomology classes in a specified subset of H2(X\D,R) containing the compactly supported Kähler classes. By imposing the condition h2, 0(X) = 0 on X, we can ensure that X\D contains an AC Calabi-Yau metric in every cohomology class in H2(X\D,R) that can be represented by a positive (1, 1)-form. This gives rise to new families of Ricci-flat Kähler metrics on certain non-compact Kähler manifolds. We furthermore construct AC Calabi-Yau metrics on smoothings of certain Calabi- Yau cones whose underlying complex space can be described by a complete intersection. As a consequence of the rate of convergence of these metrics to their asymptotic cone, we deduce from a theorem of Chan that any singular compact Calabi-Yau 3-fold with singularities modelled on the cubic [equation not reproduced here - see pdf of thesis], or on the complete intersection of two quadric cones in C⁵, both endowed with appropriate Ricci-flat metrics, admits a deformation. This last result is consistent with work of Gross.510Imperial College Londonhttp://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.540652http://hdl.handle.net/10044/1/8991Electronic Thesis or Dissertation
collection NDLTD
sources NDLTD
topic 510
spellingShingle 510
Conlon, Ronan Joseph
On the construction of asymptotically conical Calabi-Yau manifolds
description This thesis is concerned with the construction of asymptotically conical (AC) Calabi-Yau manifolds. We provide an alternative proof of a result by Goto that states that the basic (p, 0)-Hodge numbers of a positive Sasaki manifold vanish for p > 0. Our main theorem then gives sufficient conditions on a non-compact Kähler manifold to admit an AC Calabi-Yau metric in each compactly supported Kähler class. As a corollary to this, we recover a result of van Coevering which guarantees the existence of an AC Calabi-Yau metric in each compactly supported Kähler class of a crepant resolution of a Calabi-Yau cone. It also follows that we are able to give sufficient conditions on a pair (X, D) , where X is a compact Kähler manifold and D is a divisor supporting the anti-canonical bundle of X, for X\D to admit an AC Calabi-Yau metric in each compactly supported Kähler class. We extend this last result to include cohomology classes in a specified subset of H2(X\D,R) containing the compactly supported Kähler classes. By imposing the condition h2, 0(X) = 0 on X, we can ensure that X\D contains an AC Calabi-Yau metric in every cohomology class in H2(X\D,R) that can be represented by a positive (1, 1)-form. This gives rise to new families of Ricci-flat Kähler metrics on certain non-compact Kähler manifolds. We furthermore construct AC Calabi-Yau metrics on smoothings of certain Calabi- Yau cones whose underlying complex space can be described by a complete intersection. As a consequence of the rate of convergence of these metrics to their asymptotic cone, we deduce from a theorem of Chan that any singular compact Calabi-Yau 3-fold with singularities modelled on the cubic [equation not reproduced here - see pdf of thesis], or on the complete intersection of two quadric cones in C⁵, both endowed with appropriate Ricci-flat metrics, admits a deformation. This last result is consistent with work of Gross.
author2 Haskins, Mark
author_facet Haskins, Mark
Conlon, Ronan Joseph
author Conlon, Ronan Joseph
author_sort Conlon, Ronan Joseph
title On the construction of asymptotically conical Calabi-Yau manifolds
title_short On the construction of asymptotically conical Calabi-Yau manifolds
title_full On the construction of asymptotically conical Calabi-Yau manifolds
title_fullStr On the construction of asymptotically conical Calabi-Yau manifolds
title_full_unstemmed On the construction of asymptotically conical Calabi-Yau manifolds
title_sort on the construction of asymptotically conical calabi-yau manifolds
publisher Imperial College London
publishDate 2011
url http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.540652
work_keys_str_mv AT conlonronanjoseph ontheconstructionofasymptoticallyconicalcalabiyaumanifolds
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