Kai Behrend

In this dissertation, the main objects of study are sheaves with a 'twisted endomorphism' on surfaces with an automorphism. The results are divided into three main chapters:In Chapter 2, we work generally and define these objects on varieties with an automorphism. The moduli stacks of these objects are shown to be algebraic. When the automorphism is of finite order, we show that the category of such objects is equivalent to a category of modules over a certain sheaf of noncommutative algebras. After fixing a stability condition, we study Hilbert schemes of points of this sheaf of noncommutative algebras.In Chapter 3, we specialize to the case of surfaces. The topological Euler characteristics of the Hilbert schemes in Chapter 2 are packaged in a generating series, which we compute. The answer generalizes the situation when the automorphism is the identity.Chapter 4 concerns virtual counts, or Donaldson-Thomas (DT) type invariants arising from these objects. We define DT-type invariants when the structure sheaf of the surface has no higher cohomology and the quotient map is a cyclic covering of smooth surfaces. This gives new examples of invariants for automorphisms of some del Pezzo surfaces, which also may be thought of as invariants of noncommutative Fano threefolds.

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This dissertation demonstrates the utility of homotopical ideas arising from derived geometry and homotopical Lie algebras (L-infinity algebras), by describing how one can apply finite dimensional Batalin-Vilkovisky formalism to the study of Chern-Simons theory. In 1989, Witten famously demonstrated that one can use quantum Chern-Simons theory to produce topological invariants. Unfortunately, many of the ideas behind quantum gauge theory remain enigmatic to this day. In particular, the mathematically notorious path integral has yet to be fully understood. BV formalism provides a framework for understanding perturbative path integrals in a mathematically rigorous manner. However, with BV formalism, one often has to assume that the gauge theory contexts are finite dimensional. It is by now a matter of conventional wisdom, that deformation theoretic problems in characteristic zero are "governed" by L-infinity algebras. The conventional L-infinity governing the deformation theory of flat connections on a 3-manifold, which are the objects of study in Chern-Simons theory, is infinite dimensional on the chain level but has finite dimensional cohomology. By triangulating a 3-manifold, one can produce a finite-dimensional homotopically equivalent L-infinity algebra (the Whitney L-infinity algebra) that also describes the moduli of flat connections. In this thesis, we study geometric L-infinity algebras that are homotopic to ones present in conventional BV formalism. We refer to these as BV L-infinity algebras. A BV L-infinity algebra describes a certain analytic space X that can locally be embedded in some ambient smooth space as a critical locus. Although the embeddings may not assemble in a way that describes X as a global critical locus, the different embeddings can be seen being "homotopically coherent" with each other. We show that the ideas from BV formalism can be extended this homotopical setting. When equipped with an "orientation", one can produce perturbative path integral invariants in this homotopical setting.

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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 = . 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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