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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.