On the Construction of Asymptotically Conical Calabi-Yau manifolds
Author(s)
Conlon, Ronan Joseph
Type
Thesis
Abstract
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.
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.
Date Issued
2011
Date Awarded
2011-10
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Haskins, Mark
Sponsor
EPSRC
Creator
Conlon, Ronan Joseph
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)