Comptes Rendus
Algebraic Geometry
Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry
[Invariants des variétés symplectiques rationnelles réelles de dimension quatre, et bornes inférieures en géométrie énumérative réelle]
Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 341-344.

Suivant l'approche de Gromov et Witten, nous construisons des invariants par déformation des variétés symplectiques réelles rationnelles de dimension quatre. Ces invariants fournissent des bornes inférieures pour le nombre de courbes J-holomorphes rationnelles réelles de classe d'homologie donnée passant par une configuration réelle de points donnée.

Following the approach of Gromov and Witten, we construct invariants under deformation of real rational symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational J-holomorphic curves in a given homology class passing through a given real configuration of points.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00059-1

Jean-Yves Welschinger 1

1 École normale supérieure de Lyon, Unité de mathématiques pures et appliquées, 46, allée d'Italie, 69364, Lyon cedex 07, France
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Jean-Yves Welschinger. Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry. Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 341-344. doi : 10.1016/S1631-073X(03)00059-1. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00059-1/

[1] A.I. Degtyarev; V.M. Kharlamov Topological properties of real algebraic varieties: Rokhlin's way, Uspekhi Mat. Nauk, Volume 55 (2000), pp. 129-212

[2] H. Hofer; V. Lizan; J.-C. Sikorav On genericity for holomorphic curves in four-dimensional almost-complex manifolds, J. Geom. Anal., Volume 7 (1997), pp. 149-159

[3] S. Ivashkovich; V.V. Shevchishin Structure of the moduli space in a neighborhood of a cusp-curve and meromorphic hulls, Invent. Math., Volume 136 (1999), pp. 571-602

[4] M. Kontsevich; Yu. Manin Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., Volume 164 (1994), pp. 525-562

[5] D. McDuff; D. Salamon A survey of symplectic 4-manifolds with b+=1, Turkish J. Math., Volume 20 (1996), pp. 47-60

[6] V.V. Shevchishin, Pseudoholomorphic curves and the symplectic isotopy problem, Preprint , 2000 | arXiv

[7] F. Sottile, Enumerative real algebraic geometry, electronic survey at http://www.maths.univ-rennes1.fr/~raag01/surveys/ERAG/index.html, 2002

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  • Michel van Garrel; D. Peter Overholser; Helge Ruddat Enumerative Aspects of the Gross-Siebert Program, Calabi-Yau Varieties: Arithmetic, Geometry and Physics, Volume 34 (2015), p. 337 | DOI:10.1007/978-1-4939-2830-9_11
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  • Jonathan D. Hauenstein; Frank Sottile Algorithm 921, ACM Transactions on Mathematical Software, Volume 38 (2012) no. 4, p. 1 | DOI:10.1145/2331130.2331136
  • Luis D. García-Puente; Nickolas Hein; Christopher Hillar; Abraham Martín del Campo; James Ruffo; Frank Sottile; Zach Teitler The Secant Conjecture in the Real Schubert Calculus, Experimental Mathematics, Volume 21 (2012) no. 3, p. 252 | DOI:10.1080/10586458.2012.661323
  • Monique Azar; Andrei Gabrielov Some Lower Bounds in the B. and M. Shapiro Conjecture for Flag Varieties, Discrete Computational Geometry, Volume 46 (2011) no. 4, p. 636 | DOI:10.1007/s00454-010-9314-8
  • A. Arroyo; E. Brugalle; L. L. d. Medrano Recursive Formulas for Welschinger Invariants of the Projective Plane, International Mathematics Research Notices (2010) | DOI:10.1093/imrn/rnq096
  • Séverine Fiedler-Le Touzé Pencils of cubics as tools to solve an interpolation problem, Applicable Algebra in Engineering, Communication and Computing, Volume 18 (2007) no. 1-2, p. 53 | DOI:10.1007/s00200-006-0028-3
  • I. Itenberg; V. Kharlamov; E. Shustin New cases of logarithmic equivalence of Welschinger and Gromov-Witten invariants, Proceedings of the Steklov Institute of Mathematics, Volume 258 (2007) no. 1, p. 65 | DOI:10.1134/s0081543807030078
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  • Jean-Yves Welschinger Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Inventiones mathematicae, Volume 162 (2005) no. 1, p. 195 | DOI:10.1007/s00222-005-0445-0
  • Grigory Mikhalkin Enumerative tropical algebraic geometry in ℝ², Journal of the American Mathematical Society, Volume 18 (2005) no. 2, p. 313 | DOI:10.1090/s0894-0347-05-00477-7
  • Илья Владимирович Итенберг; Ilia Vladimirovich Itenberg; Вячеслав Михайлович Харламов; Viatcheslav Mikhailovich Kharlamov; Евгений Исаакович Шустин; Evgenii Isaakovich Shustin Логарифмическая эквивалентность инвариантов Вельшенже и Громова - Виттена, Успехи математических наук, Volume 59 (2004) no. 6, p. 85 | DOI:10.4213/rm797

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