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Abstract

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.