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UBC Theses and Dissertations
Study of Calabi-Yau geometry Kanazawa, Atsushi
Abstract
This thesis studies various aspects of Calabi-Yau manifolds and related geometry. It is organized into 6 chapters. Chapter 1 is the introduction of the thesis. It is devoted to background materials on K3 surfaces and Calabi-Yau threefolds. This chapter also serves to set conventions and notations. Chapter 2 studies the trilinear intersection forms and Chern classes of Calabi-Yau threefolds. It is concerned with an old question of Wilson. We demonstrate some numerical relations between the trilinear forms and Chern classes. Chapter 3 provides the full classification of Calabi-Yau threefolds with infinite fundamental group, based on Oguiso and Sakurai's work. Such Calabi-Yau threefolds are classified into two types: type A and type K. Chapter 4 investigates Calabi-Yau threefolds of type K from the viewpoint of mirror symmetry, namely Yukawa couplings and Strominger-Yau-Zaslow conjecture. We obtain several results parallel to what is known for Borcea-Voisin threefolds: Voisin's work on Yukawa couplings, and Gross and Wilson's work on special Lagrangian fibrations. Chapter 5 studies some non-commutative projective Calabi-Yau schemes. The aim of this chapter is twofold: to construct the first examples of non-commutative projective Calabi-Yau schemes, in the sense of Artin and Zhang, and to introduce a virtual counting theory of stable modules on them. Chapter 6 is the conclusion of this thesis. We recapitulate the results obtained in this thesis and also discuss future research directions.
Item Metadata
Title |
Study of Calabi-Yau geometry
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2014
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Description |
This thesis studies various aspects of Calabi-Yau manifolds and related geometry. It is organized into 6 chapters. Chapter 1 is the introduction of the thesis. It is devoted to background materials on K3 surfaces and Calabi-Yau threefolds. This chapter also serves to set conventions and notations. Chapter 2 studies the trilinear intersection forms and Chern classes of Calabi-Yau threefolds. It is concerned with an old question of Wilson. We demonstrate some numerical relations between the trilinear forms and Chern classes. Chapter 3 provides the full classification of Calabi-Yau threefolds with infinite fundamental group, based on Oguiso and Sakurai's work. Such Calabi-Yau threefolds are classified into two types: type A and type K. Chapter 4 investigates Calabi-Yau threefolds of type K from the viewpoint of mirror symmetry, namely Yukawa couplings and Strominger-Yau-Zaslow conjecture. We obtain several results parallel to what is known for Borcea-Voisin threefolds: Voisin's work on Yukawa couplings, and Gross and Wilson's work on special Lagrangian fibrations. Chapter 5 studies some non-commutative projective Calabi-Yau schemes. The aim of this chapter is twofold: to construct the first examples of non-commutative projective Calabi-Yau schemes, in the sense of Artin and Zhang, and to introduce a virtual counting theory of stable modules on them. Chapter 6 is the conclusion of this thesis. We recapitulate the results obtained in this thesis and also discuss future research directions.
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Genre | |
Type | |
Language |
eng
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Date Available |
2014-04-17
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivs 2.5 Canada
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DOI |
10.14288/1.0167400
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2014-05
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivs 2.5 Canada