Localization of virtual classes.(English)Zbl 0953.14035
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 [seeK. Behrend andB. 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 [seeM. F. Atiyah andR. Bott, Topology 23, 1-28 (1984;Zbl 0521.58025)]. Crucial for the proof are results fromA. 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 [M. Bershadsky, S. Cecotti, H. Ooguri andC. 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.
The localization formula is shown for the virtual fundamental class of an algebraic scheme \(X\) with a \(\mathbb C^\ast\)-equivariant perfect obstruction theory [seeK. Behrend andB. 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 [seeM. F. Atiyah andR. Bott, Topology 23, 1-28 (1984;Zbl 0521.58025)]. Crucial for the proof are results fromA. 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 [M. Bershadsky, S. Cecotti, H. Ooguri andC. 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.
Reviewer: Bernd Kreußler (Kaiserslautern)
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 14L30 | Group actions on varieties or schemes (quotients) |
| 14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
