# Localization of virtual classes

@article{Graber1997LocalizationOV, title={Localization of virtual classes}, author={Tom Graber and Rahul Pandharipande}, journal={Inventiones mathematicae}, year={1997}, volume={135}, pages={487-518} }

We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of projective space are expressed as graph sums of tautological integrals over moduli spaces of stable pointed curves (generalizing Kontsevich's genus 0 formulas). Also, excess integrals over spaces of higher genus multiple covers are computed.

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#### References

SHOWING 1-10 OF 35 REFERENCES

Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

- Mathematics
- 1996

We introduce a method of constructing the virtual cycle of any scheme associated with a tangent-obstruction complex. We apply this method to constructing the virtual moduli cycle of the moduli of… Expand

Generating Functions in Algebraic Geometry and Sums Over Trees

- Mathematics
- 1995

We calculate generating functions for the Poincare polynomials of moduli spaces of pointed curves of genus zero and of Configuration Spaces of Fulton and MacPherson. We also prove that contributions… Expand

Localization in equivariant intersection theory and the Bott residue formula

- Mathematics
- 1995

We prove the localization theorem for torus actions in equivariant intersection theory. Using the theorem we give another proof of the Bott residue formula for Chern numbers of bundles on smooth… Expand

Gromov-Witten classes, quantum cohomology, and enumerative geometry

- Physics, Mathematics
- 1994

The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic… Expand

Gromov-Witten invariants in algebraic geometry

- Mathematics
- 1997

Gromov-Witten invariants for arbitrary non-singular projective vari-eties and arbitrary genus are constructed using the techniques from [K. Behrend, B. Fantechi. The Intrinsic Normal Cone.]

Counting plane curves of any genus

- Mathematics
- 1996

We obtain a recursive formula answering the following question: How many irreducible, plane curves of degree d and (geometric) genus g pass through 3d-1+g general points in the plane? The formula is… Expand

Holomorphic anomalies in topological field theories

- Physics
- 1993

We study the stringy genus-one partition function of N = 2 SCFTs. It is shown how to compute this using an anomaly in decoupling of BRST trivial states from the partition function. A particular limit… Expand

Notes on stable maps and quantum cohomology

- Mathematics
- 1996

These are notes from a jointly taught class at the University of Chicago and lectures by the first author in Santa Cruz. Topics covered include: construction of moduli spaces of stable maps,… Expand

Equivariant intersection theory

- Mathematics
- 1996

In this paper we develop an equivariant intersection theory for actions of algebraic groups on algebraic schemes. The theory is based on our construction of equivariant Chow groups. They are… Expand

The intrinsic normal cone

- Mathematics
- 1997

Abstract.Let
$X$ be an algebraic stack in the sense of Deligne-Mumford. We construct a purely
$0$-dimensional algebraic stack over
$X$ (in the sense of Artin), the intrinsic normal cone
${\frak… Expand