## Duke Mathematical Journal

### Topological sigma model and Donaldson-type invariants in Gromov theory

Yongbin Ruan

#### Article information

Source
Duke Math. J. Volume 83, Number 2 (1996), 461-500.

Dates
First available in Project Euclid: 19 February 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1077244452

Digital Object Identifier
doi:10.1215/S0012-7094-96-08316-7

Mathematical Reviews number (MathSciNet)
MR1390655

Zentralblatt MATH identifier
0864.53032

#### Citation

Ruan, Yongbin. Topological sigma model and Donaldson-type invariants in Gromov theory. Duke Math. J. 83 (1996), no. 2, 461--500. doi:10.1215/S0012-7094-96-08316-7. http://projecteuclid.org/euclid.dmj/1077244452.

#### References

• [B] K. Brown, Cohomology of Groups, Grad. Texts in Math., vol. 87, Springer-Verlag, New York, 1982.
• [CF] P. Conner and F. Floyd, Differentiable Periodic Maps, Ergeb. Math. Grenzgeb., vol. 33, Academic Press, New York, 1964.
• [DG] I. Dolgachev and M. Gross, Elliptic threefolds I, Ogg-Shafarevich theory, J. Algebraic Geom. 3 (1994), no. 1, 39–80.
• [DP] S. K. Donaldson and P. Kronheimer, The Geometry of Four-Manifolds, Oxford Math. Monographs, Oxford Univ. Press, New York, 1990.
• [F] A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), no. 4, 575–611.
• [G] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347.
• [Gr] M. Gross, private communication.
• [Gr1] M. Gross, The deformation space of Calabi-Yau $n$-folds with canonical singularities can be obstructed, preprint.
• [M1] D. McDuff, Examples of symplectic structures, Invent. Math. 89 (1987), no. 1, 13–36.
• [M2] D. McDuff, Elliptic methods in symplectic geometry, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 311–358.
• [M3] D. McDuff, The structure of rational and ruled symplectic $4$-manifolds, J. Amer. Math. Soc. 3 (1990), no. 3, 679–712.
• [M4] D. McDuff, Lectures on symplectic $4$-manifolds, CIMPA Summer School, Nice 1992, preprint.
• [M5] D. McDuff, Symplectic manifolds with contact type boundaries, Invent. Math. 103 (1991), no. 3, 651–671.
• [MS] D. McDuff and D. Salamon, $J$-holomorphic curves and quantum cohomology, University Lecture Series, vol. 6, American Mathematical Society, Providence, RI, 1994.
• [Mi] J. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980.
• [Mr] D. Morrison, Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians, J. Amer. Math. Soc. 6 (1993), no. 1, 223–247.
• [O] T. Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988.
• [PW] T. Parker and J. Wolfson, A compactness theorem for Gromov's moduli space, preprint.
• [R] M. Reid, Decomposition of toric morphisms, Arithmetic and Geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser, Boston, 1983, pp. 395–418.
• [R1] Y. Ruan, Symplectic topology on algebraic $3$-folds, J. Differential Geom. 39 (1994), no. 1, 215–227.
• [R2] Y. Ruan, Symplectic topology and extremal rays, Geom. Funct. Anal. 3 (1993), no. 4, 395–430.
• [R3] Y. Ruan, Symplectic topology and complex surfaces, preprint, Mathematical Sciences Research Institute.
• [RT] Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, Math. Res. Lett. 1 (1994), no. 2, 269–278.
• [V] C. Vafa, Topological mirrors and quantum rings, Essays on Mirror Manifolds, International Press, Hong Kong, 1992, pp. 96–119.
• [Wi] E. Witten, Topological sigma models, Comm. Math. Phys. 118 (1988), no. 3, 411–449.
• [W1] P. Wilson, The Kähler cone on Calabi-Yau threefolds, Invent. Math. 107 (1992), no. 3, 561–583.
• [Wo] J. Wolfson, Gromov's compactness of pseudo-holomorphic curves and symplectic geometry, J. Differential Geom. 28 (1988), no. 3, 383–405.
• [Y] R. Ye, Gromov's compactness theorem for pseudo-holomorphic curves, Trans. Amer. Math. Soc. 342 (1994), no. 2, 671–694.