Gromovin–Wittenin teoria ja virtuaalinen lokalisaatio

Gromov-Witten theory is a curve counting theory in modern enumerative geometry, developed rigorously in the 1990s. The theory is based on the notion of a stable map from a curve to an ambient space and counts of such maps are obtained via intersection theory in the space parametrizing such maps. Many technical tools have been developed for computations in Gromov-Witten theory and perhaps the most important of these is the virtual localization formula. The goal of this thesis is to present the statement, proof and applications of this result. In order to apply the localization formula in Gromov-Witten theory, it must be proved for geometric objects called stacks. The theorem concerns the so-called virtual classes in the Chow group of a stack and thus in order to give a rigorous statement, one has to define three things: stacks, their Chow groups and virtual classes. The definitions are somewhat involved, mostly scattered in several original papers and often ignored in introductory material. This thesis presents the essential constructions and results of the original papers. The proof of the localization theorem is then based on the definition of virtual classes and properties of Chern classes on stacks. The rest of the thesis focuses on applications of the localization theorem in Gromov-Witten theory. Before the applications, a general introduction to Gromov-Witten theory is presented. More specifically, the stack of stable maps is constructed and some important geometric properties of stable maps are proved. The virtual localization formula is then applied to the Gromov-Witten theory of the projective line. In particular, using an explicit formula for the localization, Hurwitz numbers are expressed as integrals of tautological classes in the moduli space of curves. As another application, the explicit formula is used to evaluate so-called Hodge integrals. These examples illustrate the interesting and non-trivial consequences of the localization theorem.