Duke Mathematical Journal

Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants

Jean-Yves Welschinger

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let X be a real algebraic convex 3-manifold whose real part is equipped with a Pin structure. We show that every irreducible real rational curve with nonempty real part has a canonical spinor state belonging to {± 1}. The main result is then that the algebraic count of the number of real irreducible rational curves in a given numerical equivalence class passing through the appropriate number of points does not depend on the choice of the real configuration of points, provided that these curves are counted with respect to their spinor states. These invariants provide lower bounds for the total number of such real rational curves independently of the choice of the real configuration of points.

Article information

Duke Math. J., Volume 127, Number 1 (2005), 89-121.

First available in Project Euclid: 4 March 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14N35, 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]
Secondary: 14P25: Topology of real algebraic varieties


Welschinger, Jean-Yves. Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants. Duke Math. J. 127 (2005), no. 1, 89--121. doi:10.1215/S0012-7094-04-12713-7. https://projecteuclid.org/euclid.dmj/1109963911

Export citation


  • \lccM. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3, supp. 1 (1964), 3–38.
  • \lccW. Fulton and R. Pandharipande, “Notes on stable maps and quantum cohomology” in Algebraic Geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math. 62, Part 2, Amer. Math. Soc., Providence, 1997, 45–96.
  • \lccA. Gathmann, Gromov-Witten invariants of blow-ups, J. Algebraic Geom. 10 (2001), 399–432.
  • \lccM. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347.
  • \lccI. Itenberg, V. Kharlamov, and E. Shustin, Welschinger invariant and enumeration of real rational curves, Int. Math. Res. Not. 2003, no. 49, 2639–2653.
  • \lccR. C. Kirby and L. R. Taylor, “$\Pin$ structures on low-dimensional manifolds” in Geometry of Low-dimensional Manifolds, 2 (Durham, England, 1989), London Math. Soc. Lecture Note Ser. 151, Cambridge Univ. Press, Cambridge, 1990, 177–242.
  • \lccJ. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin, 1996.
  • \lccM. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525–562.
  • \lccS. Kwon, Real aspects of the moduli space of genus zero stable maps, preprint, math.AG/0305128
  • \lccJ. W. Milnor and J. D. Stasheff, Characteristic Classes, Ann. of Math. Stud. 76, Princeton Univ. Press, Princeton, 1974.
  • \lccC. Okonek, M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces, Progr. Math. 3, Birkhäuser, Boston, 1980.
  • \lccY. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), 259–367.
  • \lccF. Sottile, “Enumerative real algebraic geometry” in Algorithmic and Quantitative Real Algebraic Geometry (Piscataway, NJ, 2001), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 60, Amer. Math. Soc., Providence, 2003, 139–179.
  • \lccJ.-Y. Welschinger, Invariants of real rational symplectic $4$-manifolds and lower bounds in real enumerative geometry, C. R. Math. Acad. Sci. Paris 336 (2003), 341–344.
  • ––––, Invariants of real symplectic $4$-manifolds and lower bounds in real enumerative geometry, preprint, 2003, to appear in Invent. Math.
  • \lccE. Witten, “Two-dimensional gravity and intersection theory on moduli space” in Surveys in Differential Geometry (Cambridge, Mass., 1990), Lehigh Univ. Press, Bethlehem, Penn., 1991, 243–310.