DOI QR코드

DOI QR Code

COUNTING REAL J-HOLOMORPHIC DISCS AND SPHERES IN DIMENSION FOUR AND SIX

  • Cho, Cheol-Hyun
  • Published : 2008.09.30

Abstract

We provide another proof that the signed count of the real J-holomorphic spheres (or J- holomorphic discs) passing through a generic real configuration of k points is independent of the choice of the real configuration and the choice of J, if the dimension of the Lagrangian submanifold L (fixed point set of involution) is two or three, and also if we assume L is orient able and relatively spin. We also assume that M is strongly semi-positive. This theorem was first proved by Welschinger in a more general setting, and we provide more natural approach using the signed degree of an evaluation map.

Keywords

holomorphic discs;anti-symplectic involution;Welschinger invariants

References

  1. C.-H. Cho, Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus, Int. Math. Res. Not. (2004), no. 35, 1803-1843
  2. C.-H. Cho, Counting real J-holomorphic discs and spheres in dimension four and six, preprint: arXiv:math.SG/0604501
  3. K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Intersection Floer Theory - Anomaly and Obstruction, 2000, Preprint
  4. V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974
  5. S.-C. Kwon, Transversality properties on the moduli space of genus 0 stable maps to a smooth rational projective surface and their real enumerative implications, preprint: arXiv:math.AG/0410379
  6. D. McDuff and D. Salamon, J-holomorphic Curves and Quantum Cohomology, University Lecture Series, 6. American Mathematical Society, Providence, RI, 1994
  7. J. Solomon, Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions, Preprint: arXiv:math.SG/0606429
  8. O. Viro, introduction to topology of real algebraic varieties, article available at http://www.math.uu.se/-oleg/es/index.html
  9. J.-Y. Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195-234 https://doi.org/10.1007/s00222-005-0445-0
  10. J.-Y. Welschinger, Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants, Duke Math. J. 127 (2005), no. 1, 89-121 https://doi.org/10.1215/S0012-7094-04-12713-7
  11. J.-Y. Welschinger, Enumerative invariants of strongly semipositive real symplectic manifolds, preprint: arXiv:math.AG/0509121
  12. D.-S. Kwon and Y.-G. Oh, Structure of the image of (pseudo)-holomorphic discs with totally real boundary condition, Appendix 1 by Jean-Pierre Rosay, Comm. Anal. Geom. 8 (2000), no. 1, 31-82 https://doi.org/10.4310/CAG.2000.v8.n1.a2

Cited by

  1. Open Gromov–Witten invariants in dimension six vol.356, pp.3, 2013, https://doi.org/10.1007/s00208-012-0883-0
  2. Real orientations, real Gromov-Witten theory, and real enumerative geometry vol.24, pp.0, 2017, https://doi.org/10.3934/era.2017.24.010
  3. Localization Computation of One-Point Disk Invariants of Projective Calabi–Yau Complete Intersections vol.332, pp.2, 2014, https://doi.org/10.1007/s00220-014-2066-1
  4. Open Gromov–Witten disk invariants in the presence of an anti-symplectic involution vol.301, 2016, https://doi.org/10.1016/j.aim.2016.06.009
  5. A recursion for counts of real curves in ℂℙ2n−1: Another proof vol.29, pp.04, 2018, https://doi.org/10.1142/S0129167X18500271