Functoriality and the Moduli of Sections, With Applications to Quasimaps

Abstract

Motivated by Gromov-Witten theory, this thesis is about moduli of maps from curves to algebraic stacks, the obstruction theories of those moduli, and the functoriality of the stacks and their obstruction theories. The first part discusses the moduli of sections S of a map Z → C from an artin stack Z to a family of twisted curves C over a base algebraic stack. The existence and basic properties of S are due to Hall-Rydh; the new result in this thesis is that S has a canonical obstruction theory (not necessarily perfect), generalizing known constructions on Deligne-Mumford substacks of S. We also work out basic functoriality properties of S and its obstruction theory. The second part proves an abelianization formula for the quasimap I-function. That is, if Z is an affine l.c.i. variety with an action by a complex reductive group G such that the quotient Z//θG is a smooth projective variety, we relate the quasimap I-functions of Z//θG and Z//θ T where T is a maximal torus of G. With the mirror theorems of Ciocane-Fontantine and Kim, this computes the genus-zero Gromov-Witten invariants of Z//θG in good cases.