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.
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Jean-Yves Welschinger 1
@article{CRMATH_2007__344_5_313_0,
author = {Jean-Yves Welschinger},
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},
}
TY - JOUR AU - Jean-Yves Welschinger 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 DO - 10.1016/j.crma.2007.01.005 LA - en ID - CRMATH_2007__344_5_313_0 ER -
Jean-Yves Welschinger. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.01.005/
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