## Duke Mathematical Journal

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

Jean-Yves Welschinger

#### Abstract

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

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

Dates
First available in Project Euclid: 4 March 2005

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1109963911

Digital Object Identifier
doi:10.1215/S0012-7094-04-12713-7

Mathematical Reviews number (MathSciNet)
MR2126497

Zentralblatt MATH identifier
1084.14056

#### Citation

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

#### References

• \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.