Kai Behrend

In this thesis, we study non-commutative projective schemes whose associated graded algebras are finite over their centers. We construct symmetric obstruction theories for their moduli spaces of stable sheaves in the Calabi–Yau-3 case. This allows us to define Donaldson–Thomas (DT) type deformation invariants. As an application, we study the quantum Fermat quintic threefold which is the quintic threefold in a quantum projective space. We give an explicit description of its local models in terms of quivers with potential. We then give a full computation of its degree zero DT invariants.

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This dissertation studies stability of 3-dimensional quadratic AS-regular algebras and their moduli. A quadratic algebra defined by a regular triple (E, L, σ) is stable if there is no node or line component of E fixed by σ. We first prove stability of the twisted homogeneous coordinate ring B(E, L, σ), then lift stability to that of A(E, L, σ) by analyzing the central element c₃ where B = A/(c₃). We study a coarse moduli space for each type, A, B, E, H, S. S-equivalence of strictly semistable algebras is studied. We compute automorphisms of AS-regular algebras and of those that appear in the boundary of the moduli. We found complete DM-stacks for 2,3-truncated algebras. Type B algebra as Zhang twist of type A is studied. We found exceptional algebras which appear in the exceptional divisor of a blowing-up at a degenerate algebra in the moduli of 3-truncations. 2-unstable algebras are also studied.

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We view the inertia construction of algebraic stacks as an operator on the Grothendieck groups of various categories of algebraic stacks. We are interested in showing that the inertia operator is (locally finite and) diagonalizable over for instance the field of rational functions of the motivic class of the affine line q = [A¹]. This is proved for the Grothendieck group of Deligne-Mumford stacks and the category of quasi-split Artin stacks. Motivated by the quasi-splitness condition we then develop a theory of linear algebraic stacks and algebroids, and define a space of stack functions over a linear algebraic stack. We prove diagonalization of the semisimple inertia for the space of stack functions. A different family of operators is then defined that are closely related to the semisimple inertia. These operators are diagonalizable on the Grothendieck ring itself (i.e. without inverting polynomials in q) and their corresponding eigenvalue decompositions are used to define a graded structure on the Grothendieck ring. We then define the structure of a Hall algebra on the space of stack functions. The commutative and non-commutative products of the Hall algebra respect the graded structure defined above. Moreover, the two multiplications coincide on the associated graded algebra. This result provides a geometric way of defining a Lie subalgebra of virtually indecomposables. Finally, for any algebroid, an ε-element is defined and shown to be contained in the space of virtually indecomposables. This is a new approach to the theory of generalized Donaldson-Thomas invariants.

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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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