Localization of virtual classes (Q1283488)

In this paper a localization formula for virtual fundamental classes is established. The interest in such a formula comes from applications to enumerative geometry. For example, it can be used to deduce formulas and algorithms which compute Gromov-Witten invariants. The localization formula is shown for the virtual fundamental class of an algebraic scheme \(X\) with a \(\mathbb C^\ast\)-equivariant perfect obstruction theory [see \textit{K. Behrend} and \textit{B. Fantechi} , Invent. Math. 128, No. 1, 45-88 (1997; Zbl 0909.14006)] under the additional assumption that there exists a \(\mathbb C^\ast\)-equivariant embedding of \(X\) in a non-singular variety. The proof deduces the localization formula for \(X\) from the well known localization formula for the smooth ambient variety [see \textit{M. F. Atiyah} and \textit{R. Bott}, Topology 23, 1-28 (1984; Zbl 0521.58025)]. Crucial for the proof are results from \textit{A. Vistoli}'s paper [Invent. Math. 97, No. 3, 613-670 (1989; Zbl 0694.14001)]. In an appendix, the formula is extended to the case of Kontsevich's moduli stacks of stable maps \(\overline{M}_{g,n}(V,\beta)\). This is applied in the paper to deduce an explicit graph summation formula for the Gromov-Witten invariants for all genera of \(\mathbb P^r\). The authors use it also to prove that the contribution of degree \(d\) covers of a fixed rational curve to the genus one Gromov-Witten invariants of a Calabi-Yau 3-fold is \(1/12d\), as predicted by physicists [\textit{M. Bershadsky, S. Cecotti, H. Ooguri} and \textit{C. Vafa}, Nucl. Phys. B 405, No. 2-3, 279-304 (1993)]. In this nicely written paper, the reader will find further consequences of the localization formula in Gromov-Witten theory.