## Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

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- by Jun Li and Gang Tian PDF
- J. Amer. Math. Soc.
**11**(1998), 119-174 Request permission

## Abstract:

In this paper, we define the virtual moduli cycle of moduli spaces with perfect tangent-obstruction theory. The two interesting moduli spaces of this type are moduli spaces of vector bundles over surfaces and moduli spaces of stable morphisms from curves to projective varieties. As an application, we define the Gromov-Witten invariants of smooth projective varieties and prove all its basic properties.## References

- V. Alexeev,
*Moduli spaces $M_{g,n}(W)$ for surfaces*, preprint. - I. V. Artamkin,
*On the deformation of sheaves*, Izv. Akad. Nauk SSSR Ser. Mat.**52**(1988), no. 3, 660–665, 672 (Russian); English transl., Math. USSR-Izv.**32**(1989), no. 3, 663–668. MR**954302**, DOI 10.1070/IM1989v032n03ABEH000805 - A. Beauville,
*Quantum cohomology of complete intersections*, preprint. - K. Behrend,
*The Gromov-Witten invariants*, Invent. Math.**127 no. 3**(1997), 601–617. - K. Behrend and B. Fantechi,
*The intrinsic normal cone*, Invent. Math.**128 no. 1**(1997), 45–88. - A. Bertram,
*Quantum Schubert Calculus*, Advance in Math.**128**(1997), 289–305. - Aaron Bertram, Georgios Daskalopoulos, and Richard Wentworth,
*Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians*, J. Amer. Math. Soc.**9**(1996), no. 2, 529–571. MR**1320154**, DOI 10.1090/S0894-0347-96-00190-7 - Ionuţ Ciocan-Fontanine,
*Quantum cohomology of flag varieties*, Internat. Math. Res. Notices**6**(1995), 263–277. MR**1344348**, DOI 10.1155/S1073792895000213 - R. Dijkgraaf, C. Faber, and G. van der Geer (eds.),
*The moduli space of curves*, Progress in Mathematics, vol. 129, Birkhäuser Boston, Inc., Boston, MA, 1995. Papers from the conference held on Texel Island, April 1994. MR**1363050**, DOI 10.1007/978-1-4612-4264-2 - P. Deligne and D. Mumford,
*The irreducibility of the space of curves of given genus*, Inst. Hautes Études Sci. Publ. Math.**36**(1969), 75–109. MR**262240**, DOI 10.1007/BF02684599 - S. K. Donaldson,
*Polynomial invariants for smooth four-manifolds*, Topology**29**(1990), no. 3, 257–315. MR**1066174**, DOI 10.1016/0040-9383(90)90001-Z - Geir Ellingsrud and Stein Arild Strømme,
*The number of twisted cubic curves on the general quintic threefold*, Math. Scand.**76**(1995), no. 1, 5–34. MR**1345086**, DOI 10.7146/math.scand.a-12522 - K. Fukaya and K. Ono,
*Arnold conjecture and Gromov-Witten invariants*, preprint (1996). - William Fulton,
*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR**732620**, DOI 10.1007/978-3-662-02421-8 - W. Fulton and R. Pandharipande,
*Notes on stable maps and quantum cohomology*, to appear in the Proceeding of AMS conference in algebraic geometry, Santa Cruz. - Phillip Griffiths and Joseph Harris,
*Principles of algebraic geometry*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR**507725** - S. Katz,
*Lectures at 1993 enumerative geometry conference in Dyrkolbotn, Norway*, private communication. - S. Katz,
*Gromov-Witten Invariants via Algebraic Geometry*, preprint. - Yujiro Kawamata,
*Unobstructed deformations. A remark on a paper of Z. Ran: “Deformations of manifolds with torsion or negative canonical bundle” [J. Algebraic Geom. 1 (1992), no. 2, 279–291; MR1144440 (93e:14015)]*, J. Algebraic Geom.**1**(1992), no. 2, 183–190. MR**1144434** - Maxim Kontsevich,
*Intersection theory on the moduli space of curves and the matrix Airy function*, Comm. Math. Phys.**147**(1992), no. 1, 1–23. MR**1171758**, DOI 10.1007/BF02099526 - Maxim Kontsevich,
*Enumeration of rational curves via torus actions*, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 335–368. MR**1363062**, DOI 10.1007/978-1-4612-4264-2_{1}2 - M. Kontsevich and Yu. Manin,
*Gromov-Witten classes, quantum cohomology, and enumerative geometry*, Comm. Math. Phys.**164**(1994), no. 3, 525–562. MR**1291244**, DOI 10.1007/BF02101490 - Finn F. Knudsen,
*The projectivity of the moduli space of stable curves. II. The stacks $M_{g,n}$*, Math. Scand.**52**(1983), no. 2, 161–199. MR**702953**, DOI 10.7146/math.scand.a-12001 - Olav Arnfinn Laudal,
*Formal moduli of algebraic structures*, Lecture Notes in Mathematics, vol. 754, Springer, Berlin, 1979. MR**551624**, DOI 10.1007/BFb0065055 - Jun Li,
*Algebraic geometric interpretation of Donaldson’s polynomial invariants*, J. Differential Geom.**37**(1993), no. 2, 417–466. MR**1205451** - J. Li and G. Tian,
*Quantum cohomology of homogeneous manifolds*, to appear in J. Alg. Geom. - J. Li and G. Tian,
*Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds*, preprint (1996). - Hideyuki Matsumura,
*Commutative ring theory*, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR**1011461** - John W. Morgan,
*Comparison of the Donaldson polynomial invariants with their algebro-geometric analogues*, Topology**32**(1993), no. 3, 449–488. MR**1231956**, DOI 10.1016/0040-9383(93)90001-C - David Mumford,
*Towards an enumerative geometry of the moduli space of curves*, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328. MR**717614** - Ziv Ran,
*Deformations of maps*, Algebraic curves and projective geometry (Trento, 1988) Lecture Notes in Math., vol. 1389, Springer, Berlin, 1989, pp. 246–253. MR**1023402**, DOI 10.1007/BFb0085936 - Yongbin Ruan,
*Topological sigma model and Donaldson-type invariants in Gromov theory*, Duke Math. J.**83**(1996), no. 2, 461–500. MR**1390655**, DOI 10.1215/S0012-7094-96-08316-7 - Y. Ruan and Z-B. Qin,
*Quantum cohomology of projective bundles over $\mathbf {P}^{n}$*, to appear in Trans. Amer. Math. Soc. - Yongbin Ruan and Gang Tian,
*A mathematical theory of quantum cohomology*, J. Differential Geom.**42**(1995), no. 2, 259–367. MR**1366548** - Yongbin Ruan and Gang Tian,
*Higher genus symplectic invariants and sigma model coupled with gravity*, Turkish J. Math.**20**(1996), no. 1, 75–83. MR**1392664** - B. Siebert,
*Gromov-Witten invariants for general symplectic manifolds*, preprint (1996). - G. Tian,
*Quantum cohomology and its associativity*, To appear in Proc. of 1st Current Developments in Math., Cambridge (1995). - Angelo Vistoli,
*Intersection theory on algebraic stacks and on their moduli spaces*, Invent. Math.**97**(1989), no. 3, 613–670. MR**1005008**, DOI 10.1007/BF01388892 - C. Voisin,
*A mathematical proof of Aspinwall-Morrison formula*, Compositio Math.**104 no. 2**(1996), 135–151. - Edward Witten,
*Topological sigma models*, Comm. Math. Phys.**118**(1988), no. 3, 411–449. MR**958805**, DOI 10.1007/BF01466725 - Edward Witten,
*Two-dimensional gravity and intersection theory on moduli space*, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 243–310. MR**1144529**

## Additional Information

**Jun Li**- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- MR Author ID: 1093262
- Email: jli@gauss.stanford.edu
**Gang Tian**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 220655
- Email: tian@math.mit.edu
- Received by editor(s): September 25, 1996
- Received by editor(s) in revised form: July 30, 1997
- Additional Notes: Both authors were supported in part by NSF grants, and the first author was also supported by an A. Sloan fellowship and Terman fellowship.
- © Copyright 1998 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**11**(1998), 119-174 - MSC (1991): Primary 14D20
- DOI: https://doi.org/10.1090/S0894-0347-98-00250-1
- MathSciNet review: 1467172