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Uspekhi Mat. Nauk, 2004, Volume 59, Issue 6(360), Pages 85–110 (Mi umn797)  

This article is cited in 28 scientific papers (total in 28 papers)

Logarithmic equivalence of Welschinger and Gromov–Witten invariants

I. V. Itenberga, V. M. Kharlamova, E. I. Shustinb

a University Louis Pasteur
b Tel Aviv University, School of Mathematical Sciences

Abstract: The Welschinger numbers, a kind of a real analogue of the Gromov–Witten numbers that count the complex rational curves through a given generic collection of points, bound from below the number of real rational curves for any generic collection of real points. Logarithmic equivalence of sequences is understood to mean the asymptotic equivalence of their logarithms. Such an equivalence is proved for the Welschinger and Gromov–Witten numbers of any toric Del Pezzo surface with its tautological real structure, in particular, of the projective plane, under the hypothesis that all, or almost all, the chosen points are real. A study is also made of the positivity of Welschinger numbers and their monotonicity with respect to the number of imaginary points.

DOI: https://doi.org/10.4213/rm797

Full text: PDF file (472 kB)
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English version:
Russian Mathematical Surveys, 2004, 59:6, 1093–1116

Bibliographic databases:

UDC: 512.77
MSC: Primary 14N10, 14N35; Secondary 53D45, 14M25, 14H15, 14J26
Received: 27.07.2004

Citation: I. V. Itenberg, V. M. Kharlamov, E. I. Shustin, “Logarithmic equivalence of Welschinger and Gromov–Witten invariants”, Uspekhi Mat. Nauk, 59:6(360) (2004), 85–110; Russian Math. Surveys, 59:6 (2004), 1093–1116

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Itenberg I., Kharlamov V., Shustin E., “Logarithmic asymptotics of the genus zero Gromov–Witten invariants of the blown up plane”, Geom. Topol., 9 (2005), 483–491  crossref  mathscinet  zmath  isi
    2. Welschinger J.-Y., “Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry”, Invent. Math., 162:1 (2005), 195–234  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. G. L. Litvinov, “The Maslov dequantization, idempotent and tropical mathematics: a brief introduction”, J. Math. Sci. (N. Y.), 140:3 (2007), 426–444  mathnet  crossref  mathscinet  zmath  elib  elib
    4. Shustin E., “A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces”, J. Algebraic Geom., 15:2 (2006), 285–322  crossref  mathscinet  zmath  isi  elib
    5. Ruffo J., Sivan Yu., Soprunova E., Sottile F., “Experimentation and conjectures in the real Schubert calculus for flag manifolds”, Experiment. Math., 15:2 (2006), 199–221  crossref  mathscinet  zmath  isi  elib
    6. Proc. Steklov Inst. Math., 258 (2007), 65–73  mathnet  crossref  mathscinet  zmath  elib
    7. Proc. Steklov Inst. Math., 258 (2007), 218–246  mathnet  crossref  mathscinet  zmath  elib
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    9. Welschinger J.-Y., “Invariant count of holomorphic disks in the cotangent bundles of the two-sphere and real projective plane”, C. R. Math. Acad. Sci. Paris, 344:5 (2007), 313–316  crossref  mathscinet  zmath  isi  elib
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    11. Katz E., “A tropical toolkit”, Expo. Math., 27:1 (2009), 1–36  crossref  mathscinet  zmath  isi  elib
    12. “Introduction to tropical geometry”, Tropical Algebraic Geometry, Oberwolfach Seminars, 35, 2009, 1  mathscinet  isi
    13. Sottile F., “Frontiers of Reality in Schubert Calculus”, Bulletin of the American Mathematical Society, 47:1 (2010), 31–71  crossref  mathscinet  zmath  isi  elib
    14. Arroyo A., Brugalle E., Lopez de Medrano L., “Recursive Formulas for Welschinger Invariants of the Projective Plane”, Int Math Res Not, 2011, no. 5, 1107–1134  mathscinet  zmath  isi  elib
    15. Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin, “Welschinger invariants of real Del Pezzo surfaces of degree ≥ 3”, Math. Ann, 2012  crossref  mathscinet  isi
    16. Garcia-Puente L.D., Hein N., Hillar Ch., del Campo A.M., Ruffo J., Sottile F., Teitler Z., “The Secant Conjecture in the Real Schubert Calculus”, Exp. Math., 21:3 (2012), 252–265  crossref  mathscinet  zmath  isi
    17. Shustin E., “Tropical and Algebraic Curves with Multiple Points”, Perspectives in Analysis, Geometry, and Topology: on the Occasion of the 60th Birthday of Oleg Viro, Progress in Mathematics, 296, eds. Itenberg I., Joricke B., Passare M., Birkhauser Verlag Ag, 2012, 431–464  crossref  mathscinet  zmath  isi
    18. Oancea A., “Welschinger's Invariants”, Asterisque, 2012, no. 348, 265–297  mathscinet  zmath  isi
    19. Itenberg I., Kharlamov V., Shustin E., “Welschinger Invariants of Small Non-Toric Del Pezzo Surfaces”, J. Eur. Math. Soc., 15:2 (2013), 539–594  crossref  mathscinet  zmath  isi  elib
    20. Brugalle E. Puignau N., “Enumeration of Real Conics and Maximal Configurations”, J. Eur. Math. Soc., 15:6 (2013), 2139–2164  crossref  mathscinet  zmath  isi
    21. Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin, “Welschinger invariants of real del Pezzo surfaces of degree ≥ 2”, Int. J. Math, 26:08 (2015), 1550060  crossref  mathscinet  zmath  isi
    22. Brugalle E., “Floor Diagrams Relative To a Conic, and Gw-W Invariants of Del Pezzo Surfaces”, 279, 2015, 438–500  crossref  mathscinet  zmath  isi
    23. Brugalle E., Puignau N., “on Welschinger Invariants of Symplectic 4-Manifolds”, 90, no. 4, 2015, 905–938  crossref  mathscinet  zmath  isi
    24. Hein N., Sottile F., Zelenko I., “A congruence modulo four in real Schubert calculus”, J. Reine Angew. Math., 714 (2016), 151–174  crossref  mathscinet  zmath  isi  elib  scopus
    25. Hein N., Sottile F., Zelenko I., “A Congruence Modulo Four For Real Schubert Calculus With Isotropic Flags”, Can. Math. Bul.-Bul. Can. Math., 60:2 (2017), 309–318  crossref  mathscinet  zmath  isi
    26. Itenberg I., Zvonkine D., “Hurwitz Numbers For Real Polynomials”, Comment. Math. Helv., 93:3 (2018), 441–474  crossref  mathscinet  zmath  isi  scopus
    27. Ding Ya., Hu J., “Welschinger Invariants of Blow-Ups of Symplectic 4-Manifolds”, Rocky Mt. J. Math., 48:4 (2018), 1105–1144  crossref  mathscinet  zmath  isi  scopus
    28. Brugalle E., “Surgery of Real Symplectic Fourfolds and Welschinger Invariants”, J. Singul., 17 (2018), 267–294  crossref  mathscinet  zmath  isi  scopus
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