Algebraic Geometry
Invariant count of holomorphic disks in the cotangent bundles of the two-sphere and real projective plane
Comptes Rendus. Mathématique, Volume 344 (2007) no. 5, pp. 313-316.

We first define enumerative invariants of the cotangent bundles of the two-sphere and real projective plane. These invariants are obtained in the framework of symplectic field theory by counting with respect to some sign holomorphic disks with punctures sitting on the zero section. Then, we relate these invariants with the ones of closed real symplectic four-manifolds which have been constructed earlier. This relation provides some congruences and recursive formulas for the latter as well as sharpness results for the associated lower bounds in real enumerative geometry.

Dans une première partie, nous introduisons des invariants énumératifs des fibrés cotangents de la sphère de dimension deux et du plan projectif réel. Ces invariants sont obtenus dans le langage de la théorie symplectique des champs en comptant en fonction d'un signe les disques holomorphes avec pointes qui reposent sur la section nulle. Puis nous relions ces invariants avec ceux des variétés symplectiques réelles de dimension quatre précédemment construits et déduisons des congruences et formules récurrentes pour ces derniers, ainsi que des résultats d'optimalité pour les bornes inférieures associées en géométrie énumérative réelle.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.01.005
Welschinger, Jean-Yves 1

1 École normale supérieure de Lyon, unité de mathématiques pures et appliquées, UMR CNRS 5669, 46, allée d'Italie, 69364 Lyon cedex 07, France
@article{CRMATH_2007__344_5_313_0,
     author = {Welschinger, Jean-Yves},
     title = {Invariant count of holomorphic disks in the cotangent bundles of the two-sphere and real projective plane},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {313--316},
     publisher = {Elsevier},
     volume = {344},
     number = {5},
     year = {2007},
     doi = {10.1016/j.crma.2007.01.005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2007.01.005/}
}
TY  - JOUR
AU  - Welschinger, Jean-Yves
TI  - Invariant count of holomorphic disks in the cotangent bundles of the two-sphere and real projective plane
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 313
EP  - 316
VL  - 344
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2007.01.005/
DO  - 10.1016/j.crma.2007.01.005
LA  - en
ID  - CRMATH_2007__344_5_313_0
ER  - 
%0 Journal Article
%A Welschinger, Jean-Yves
%T Invariant count of holomorphic disks in the cotangent bundles of the two-sphere and real projective plane
%J Comptes Rendus. Mathématique
%D 2007
%P 313-316
%V 344
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2007.01.005/
%R 10.1016/j.crma.2007.01.005
%G en
%F CRMATH_2007__344_5_313_0
Welschinger, Jean-Yves. Invariant count of holomorphic disks in the cotangent bundles of the two-sphere and real projective plane. Comptes Rendus. Mathématique, Volume 344 (2007) no. 5, pp. 313-316. doi : 10.1016/j.crma.2007.01.005. http://www.numdam.org/articles/10.1016/j.crma.2007.01.005/

[1] F. Bourgeois, A Morse–Bott approach to Contact Homology, Ph.D. dissertation, Stanford University, 2002

[2] Eliashberg, Y.; Givental, A.; Hofer, H. Introduction to symplectic field theory, GAFA 2000 (Tel Aviv, 1999) (Geom. Funct. Anal.) (2000), pp. 560-673 (Special Volume, Part II)

[3] Gathmann, A.; Markwig, H. The Caporaso–Harris formula and plane relative Gromov–Witten invariants in tropical geometry (preprint) | arXiv

[4] Hofer, H.; Wysocki, K.; Zehnder, E. Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 13 (1996) no. 3, pp. 337-379

[5] Itenberg, I.; Kharlamov, V.; Shustin, E. Logarithmic equivalence of the Welschinger and the Gromov–Witten invariants, Uspekhi Mat. Nauk, Volume 59 (2004) no. 6, pp. 85-110 (Translation in Russian Math. Surveys, 59, 6, 2004, pp. 1093-1116)

[6] Itenberg, I.; Kharlamov, V.; Shustin, E. A Caporaso–Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces, 2006 (preprint) | arXiv

[7] Mikhalkin, G. Enumerative tropical algebraic geometry in R2, J. Amer. Math. Soc., Volume 18 (2005) no. 2, pp. 313-377

[8] Shustin, E. A tropical calculation of the Welschinger invariants of real toric del Pezzo surfaces, J. Algebraic Geom., Volume 15 (2006) no. 2, pp. 285-322

[9] Vakil, R. Counting curves on rational surfaces, Manuscripta Math., Volume 102 (2000) no. 1, pp. 53-84

[10] Welschinger, J.-Y. Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math., Volume 162 (2005) no. 1, pp. 195-234

Cited by Sources: