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Pencils of quadrics and Gromov–Witten–Welschinger invariants of \(\mathbb {C}P^3\)

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Abstract

We establish a formula for the Gromov–Witten–Welschinger invariants of \(\mathbb {C}P^3\) with mixed real and conjugate point constraints. The method is based on a suggestion by J. Kollár that, considering pencils of quadrics, some real and complex enumerative invariants of \(\mathbb {C}P^3\) could be computed in terms of enumerative invariants of \(\mathbb {C}P^1\times \mathbb {C}P^1\) and of elliptic curves.

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Notes

  1. Such a choice is possible since every trivialization of \(T\mathbb {R}P^3\) over the 2-skeleton extends to the 3-skeleton. The two homotopy classes of trivializations over the 2-skeleton correspond to different values of \(s_{\mathbb {R}P^3}(D)\).

References

  1. Arroyo, A., Brugallé, E., Lopez de Medrano, L.: Recursive formula for Welschinger invariants. Int. Math. Res. Not. 5, 1107–1134 (2011)

    MATH  Google Scholar 

  2. Brugallé, E., Mikhalkin, G.: Enumeration of curves via floor diagrams. C. R. de l’Académie des Sci. de Paris série I 345(6), 329–334 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brugallé, E., Mikhalkin, G.: Floor decompositions of tropical curves: the planar case. In: Proceedings of 15th Gökova Geometry-Topology Conference, pp. 64–90 (2008)

  4. Brugallé, E.: Floor diagrams relative to a conic, and GW–W invariants of Del Pezzo surfaces. Adv. Math. 279, 438–500 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gross, B.H., Harris, J.: Real algebraic curves. Ann. Sci. École Norm. Sup. (4) 14(2), 157–182 (1981)

    MathSciNet  MATH  Google Scholar 

  6. Griffiths, P., Harris, J.: Infinitesimal variations of Hodge structure II. An infinitesimal invariant of Hodge classes. Compos. Math. 50(2–3), 207–265 (1983)

    MathSciNet  MATH  Google Scholar 

  7. Georgieva, P., Zinger, A.: Enumeration of real curves in \(\mathbb{C} P^{2n-1}\) and a WDVV relation for real Gromov–Witten invariants. arXiv:1309.4079 (2013)

  8. Horev, A., Solomon, J.: The open Gromov–Witten–Welschinger theory of blowups of the projective plane. arXiv:1210.4034 (2012)

  9. Itenberg, I., Kharlamov, V., Shustin, E.: Welschinger invariants of real Del Pezzo surfaces of degree \(\ge 2\). arXiv:1312.2921 (2013)

  10. Kollár, J.: Examples of vanishing Gromov–Witten–Welschinger invariants. J. Math. Sci. Univ. Tokyo 22(1), 261–278 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Kharlamov, Viatcheslav, Răsdeaconu, Rareş: Counting real rational curves on \(K3\) surfaces. Int. Math. Res. Not. IMRN 14, 5436–5455 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mikhalkin, G.: Enumerative tropical algebraic geometry in \(\mathbb{R}^2\). J. Am. Math. Soc. 18(2), 313–377 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Natanzon, S.M.: Klein surfaces. Uspekhi Mat. Nauk. 45(6(276)), 47–90 (1990). 189

    MathSciNet  MATH  Google Scholar 

  14. Pandharipande, R., Solomon, J., Walcher, J.: Disk enumeration on the quintic 3-fold. J. Am. Math. Soc. 21(4), 1169–1209 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Shustin, E.: A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces. J. Algebraic Geom. 15, 285–322 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Welschinger, J.Y.: Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry. Invent. Math. 162(1), 195–234 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Welschinger, J.Y.: Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants. Duke Math. J. 127(1), 89–121 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Welschinger, J.Y.: Optimalité, congruences et calculs d’invariants des variétés symplectiques réelles de dimension quatre. arXiv:0707.4317 (2007)

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Acknowledgments

We are grateful to B. Bertrand, I. Itenberg, and G. Mikhalkin for useful discussions. We also wish to thank the anonymous referee for his valuable comments on the first version of this text.

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Authors and Affiliations

  1. CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128, Palaiseau Cedex, France

    Erwan Brugallé

  2. Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Pierre et Marie Curie, 4 place Jussieu, 75252, Paris Cedex 5, France

    Penka Georgieva

Authors
  1. Erwan Brugallé
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  2. Penka Georgieva
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Correspondence to Penka Georgieva.

Additional information

P. Georgieva is supported by ERC Grant STEIN-259118.

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Brugallé, E., Georgieva, P. Pencils of quadrics and Gromov–Witten–Welschinger invariants of \(\mathbb {C}P^3\) . Math. Ann. 365, 363–380 (2016). https://doi.org/10.1007/s00208-016-1398-x

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  • DOI: https://doi.org/10.1007/s00208-016-1398-x

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