Skip to main content
SpringerLink
Log in

Algebraically unrealizable complex orientations of plane real pseudoholomorphic curves

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We prove two inequalities for the complex orientations of a separating non-singular real algebraic curve in \({\mathbb {RP}}^2\) of any odd degree. We also construct a separating non-singular real (i.e., invariant under the complex conjugation) pseudoholomorphic curve in \({\mathbb {CP}}^2\) of any degree congruent to 9 mod 12 which does not satisfy one of these inequalities. Therefore the oriented isotopy type of the real locus of each of these curves is algebraically unrealizable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. D. Auroux, S. K. Donaldson and L. Katzarkov. Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves. Math. Ann. 326(2003), 185–203

    Article  MathSciNet  Google Scholar 

  2. E. Brugallé. Symmetric plane curves of degree 7: pseudoholomorphic and algebraic classification. J. Reine Angew. Math. 612(2007), 129–171

    MathSciNet  MATH  Google Scholar 

  3. M. Coppens. The separating gonality of a separating real curve. Monatsh. Math. (1)170 (2013), 1–10

  4. T. Fiedler, Eine Beschränkung für die Lage von reellen ebenen algebraischen Kurven. Beitr. Algebra Geom. 11(1981), 7–19

    MATH  Google Scholar 

  5. S. Fiedler-LeTouzé and S. Orevkov. A flexible affine \(M\)-sextic which is algebraically unrealizable. J. Alg. Geom. 11(2002), 293–320

    Article  MathSciNet  Google Scholar 

  6. S. Fiedler-LeTouzé, S. Orevkov and E. Shustin. Corrigendum to the paper “A flexible affine M-sextic which is algebraically unrealizable”. J. Alg. Geom. 29(2020), 109–121

    Article  Google Scholar 

  7. A. Gabard. Sur la représentation conforme des surfaces de Riemann à bord et une caracérisation des courbes séparantes. Comment. Math. Helv. (4)81 (2006), 945–964

  8. M. Gromov. Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82(1985), 307–347

    Article  MathSciNet  Google Scholar 

  9. D. Hilbert. Über die reellen Züge der algebraischen Kurven. Math. Ann. 38(1891), 115–137

    Article  MathSciNet  Google Scholar 

  10. S. Ivashkovich and V. Shevchishin. Complex curves in almost complex manifolds and meromorphic hulls. arXiv:math/9912046

  11. M. Kummer and K. Shaw. The separating semigroup of a real curve. Ann. de la fac. des sci. de Toulouse. Math. (6), (1)29 (2020), 79–96

  12. A. Marin. La transversalité topologique. Une extension d’un théorème de Rohlin et application au 16e problème de Hilbert. Thèse d’Etat, Univ. Paris-Sud (1979)

  13. S. Yu. Orevkov. Quasipositivity test via unitary representations of braid groups and its applications to real algebraic curves. J. Knot Theory Ramif. 10(2001), 1005–1023

    Article  MathSciNet  Google Scholar 

  14. S. Yu. Orevkov. Classification of flexible \(M\)-curves of degree \(8\) up to isotopy. GAFA - Geom. and Funct. Anal. 12(2002), 723–755

    Article  MathSciNet  Google Scholar 

  15. S. Yu. Orevkov. Construction of arrangements of an \(M\)-quartic and an \(M\)-cubic with a maximal intersection of an oval and the odd branch, Bull. (vestnik) of Nizhni Novgorod State Univ., Ser. Math. Modeling and Optimal Control, (1)(25) (2002), 12–48 (Russian); English transl., https://www.math.univ-toulouse.fr/orevkov/j3o4e.pdf

  16. S. Yu. Orevkov. Riemann existence theorem and construction of real algebraic curves. Ann. Fac. Sci. de Toulouse. Math. Math. (6), (4)12 (2003), 517–531

  17. S. Yu. Orevkov. Arrangements of an \(M\)-quintic with respect to a conic that maximally intersects its odd branch. Algebra i Analiz (4)19 (2007), 174–242 (Russian); English transl., St. Petersburg Math. J. 19(2008), 625–674

  18. S. Yu. Orevkov. Some examples of real algebraic and real pseudoholomorphic curves. In: Perspectives in Analysis, Geometry and Topology, Progr. in Math, Vol 296. Birkhauser/Springer, N. Y. (2012), 355–387

  19. S. Yu. Orevkov. Separating semigroup of hyperelliptic curves and of genus 3 curves. Algebra i analiz (1)31 (2019), 108–113 (Russian); English transl. St. Petersburg Math. J. 31 (2020), 81–84

  20. S. Yu. Orevkov and E. I. Shustin. Flexible algebraically unrealizable curves: rehabilitation of Hilbert-Rohn-Gudkov approach. J. für die Reine und Angew. Math. 511(2002), 145–172

  21. S. Yu. Orevkov and E. I. Shustin. Pseudoholomorphic algebraically unrealizable curves. Moscow Math. J. 3(2003), 1053–1083

  22. S. Yu. Orevkov and E. I. Shustin. Real algebraic and pseudoholomorphic curves on the quadratic cone and smoothings of singularity \(X_{21}\). Algebra i Analiz (2)28 (2016), 138–186 (Russian): English transl. St. Petersburg Math. J. 28(2017), 225–257

  23. V. A. Rokhlin. Complex topological characteristics of real algebraic curves. Usp. mat. nauk (5)33 (1978), 77–89 (Russian); English transl. Russian Math. Surveys (5)33 (1978), 85–98

  24. O. Ya. Viro. Real algebraic plane curves: constructions with controlled topology. Algebra i analiz (5)1 (1989), 1–73 (Russian); English transl. Leningrad J. Math. 1(1990), 1059–1134

  25. A. Weinstein. Symplectic manifolds and their Lagrangian submanifolds. Adv. Math. 6(1971), 329–346

    Article  MathSciNet  Google Scholar 

  26. J.-Y. Welschinger. Courbes algébriques réelles et courbes flexibles sur les surfaces réglées de base \(CP^1\). Proc. London Math. Soc. (3)85 (2002), 367–392

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I am grateful to G. Mikhalkin, S. Nemirovski, and especially to E. Shustin for many stimulating discussions. To a great extent this work was inspired by Kummer and Shaw’s paper [KS20] though I did not use explicitly their results. I thank the referee for many useful remarks and suggestions.

Author information

Authors and Affiliations

  1. IMT, l’université Paul Sabatier, 118 route de Narbonne, Toulouse, France

    S. Yu. Orevkov

  2. Steklov Mathematical Institute, Gubkina 8, Moscow, Russia

    S. Yu. Orevkov

  3. AGHA Laboratory, Moscow Institute of Physics and Technology, Moscow, Russia

    S. Yu. Orevkov

Authors
  1. S. Yu. Orevkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Yu. Orevkov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Orevkov, S.Y. Algebraically unrealizable complex orientations of plane real pseudoholomorphic curves. Geom. Funct. Anal. 31, 930–947 (2021). https://doi.org/10.1007/s00039-021-00569-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-021-00569-1

Search

Navigation