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Title: On the construction of asymptotically conical Calabi-Yau manifolds
Author: Conlon, Ronan Joseph
ISNI:       0000 0004 2706 2463
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2011
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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.
Supervisor: Haskins, Mark Sponsor: EPSRC
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral