Jim Bryan

This dissertation consists of two main chapters, each pertaining to the enumerative geometry of Calabi-Yau manifolds with an action:In Chapter 2 we study Euler characteristics of the G-invariant Hilbert schemes of points on an Abelian surface with a symplectic action by finite group G. One can package these Euler characteristics into generating series, whose reciprocal we prove is a holomorphic modular form for a particular congruence subgroup. For the standard involution of multiplication by -1, we prove an analogue of the Yau-Zaslow formula--that is, these Euler characteristics determine a weighted number of curves invariant under the involution and with rational quotient. Motivated by the results of Chapter 2, in Chapter 3 we develop in more generality a theory counting invariant curves in Calabi-Yau threefolds with an involution. Our theory conjecturally results in analogues of the Gopakumar-Vafa invariants which count invariant curves of genus g and with genus h quotient. We prove the conjecture and compute all invariants in the case of a local Abelian surface with involution multiplication by -1, or a local Nikulin K3 surface together with the Nikuln involution.

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This research develops a method to compute Katz's genus 0 Gopakumar-Vafa invariants of fiber curve classes of the Banana manifold, a smooth compact Calabi-Yau threefold, and its generalizations, the multi-Banana configurations. The invariants turn out to have geometric significance, being an actual count of certain local configurations of rational curves on a related threefold. In both these cases, the partition functions for the invariants exhibit a modularity and can be expressed in terms of Jacobi forms or theta functions.

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This thesis contains two chapters which reflect the two main viewpoints of modern enumerative geometry.In chapter 1 we develop a theory for stable maps to curves with divisible ramification. For a fixed integer r>0, we show that the condition of every ramification locus being divisible by r is equivalent to the existence of an r-th root of a canonical section. We consider this condition in regards to both absolute and relative stable maps and construct natural moduli spaces in these situations. We construct an analogue of the Fantechi-Pandharipande branch morphism and when the domain curves are genus zero we construct a virtual fundamental class. This theory is anticipated to have applications to r-spin Hurwitz theory. In particular it is expected to provide a proof of the r-spin ELSV formula [SSZ'15, Conj. 1.4] when used with virtual localisation. In chapter 2 we further the study of the Donaldson-Thomas theory of the banana threefolds which were recently discovered and studied in . These are smooth proper Calabi-Yau threefolds which are fibred by Abelian surfaces such that the singular locus of a singular fibre is a non-normal toric curve known as a “banana configuration”. In the Donaldson-Thomas partition function for the rank 3 sub-lattice generated by the banana configurations is calculated. In this chapter we provide calculations with a view towards the rank 4 sub-lattice generated by a section and the banana configurations. We relate the findings to the KKV formula and present new Gopakumar-Vafa invariants for the banana threefold.

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This thesis develops a method (dimensional reduction) to compute motivic Donaldson--Thomas invariants. The method is then employed to compute these invariants in several different cases.Given a moduli scheme with a symmetric obstruction theory a Donaldson--Thomas type invariant can be defined by integrating Behrend's function over the scheme. Motivic Donaldson--Thomas theory aims to provide a more refined invariant associated to each such moduli space - a virtual motive.From the modern point of view motivic Donaldson-Thomas invariants should be defined for a three dimensional Calabi--Yau category. These categories often arise in a geometric context as the derived category of representations of a quiver with potential.Provided the potential has a linear factor we are able to reduce the problem of computing the corresponding virtual motives to a much simpler one. This includes geometric examples coming from local curves which we compute explicitly.

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In this thesis we produce a generating function for the number of hyperelliptic curves (up to translation) on a polarized Abelian surface using the crepant resolution conjecture and the Yau-Zaslow formula. We present a formula to compute these in terms of P. A. MacMahon's generalized sum-of-divisors functions, and prove that they are quasi-modular forms.

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The Donaldson-Thomas (DT) theory of a Calabi-Yau threefold X gives rise to subtle deformation invariants. They are considered to be the mathematical counterparts of BPS state counts in topological string theory compactified on X. Principles of physics indicate that the string theory of a singular Calabi-Yau threefold and that of its crepant resolution ought to be equivalent, so one might expect that the DT theory of a singular Calabi-Yau threefold ought to be equivalent to that of its crepant resolution. There is some difficulty in defining DT when X is singular, but Bryan, Cadman, and Young have (in some generality) defined DT theory in the case where X is the coarse moduli space of an orbifold χ. The crepant resolution conjecture of Bryan-Cadman-Young gives a formula determining the DT invariants of the orbifold in terms of the DT invariants of the crepant resolution. In this dissertation, we begin a program to prove the crepant resolution conjecture using Hall algebra techniques inspired by those of Bridgeland.

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