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Exponential rarefaction of real curves with many components

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Abstract

Given a positive real Hermitian holomorphic line bundle L over a smooth real projective manifold X, the space of real holomorphic sections of the bundle L d inherits for every d∈ℕ∗ a L 2-scalar product which induces a Gaussian measure. When X is a curve or a surface, we estimate the volume of the cone of real sections whose vanishing locus contains many real components. In particular, the volume of the cone of maximal real sections decreases exponentially as d grows to infinity.

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References

  1. E. Bedford, M. Lyubich, and J. Smillie, Polynomial diffeomorphisms of C 2. IV. The measure of maximal entropy and laminar currents, Invent. Math., 112 (1993), 77–125.

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Berman, B. Berndtsson, and J. Sjöstrand, A direct approach to Bergman kernel asymptotics for positive line bundles, Ark. Mat., 46 (2008), 1871–2487.

    Article  Google Scholar 

  3. T. Bouche, Convergence de la métrique de Fubini-Study d’un fibré linéaire positif, Ann. Inst. Fourier, 40 (1990), 117–130.

    MathSciNet  MATH  Google Scholar 

  4. E. Catlin, The Bergman Kernel and a theorem of Tian, Analysis and Geometry in Several Complex Variables. G. Komatsu and M. Kuranishi, eds. Birkhäuser, Boston, 1999.

    Google Scholar 

  5. H. de Thélin, Sur la laminarité de certains courants, Ann. Sci. Ecole Norm. Super., 37 (2004), 304–311.

    MATH  Google Scholar 

  6. J.-P. Demailly, Courants positifs extrêmaux et conjecture de Hodge, Invent. Math., 69 (1982), 347–374.

    Article  MathSciNet  MATH  Google Scholar 

  7. T.-C. Dinh and N. Sibony, Distribution des valeurs de transformations méromorphes et applications, Comment. Math. Helv., 81 (2006), 221–258.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differ. Geom., 44 (1996), 666–705.

    MathSciNet  MATH  Google Scholar 

  9. R. Dujardin, Laminar currents in \({\Bbb{P}}^{2}\), Math. Ann., 325 (2003), 745–765.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Edelman and E. Kostlan, How many zeros of a random polynomial are real?, Bull. Am. Math. Soc., 32 (1995), 1–37.

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Gayet, Symplectic real hypersurfaces and Lefschetz pencils, J. Symplectic Geom., 6 (2008), 247–266.

    MathSciNet  MATH  Google Scholar 

  12. A. Harnack, Ueber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann., 10 (1876), 189–198.

    Article  MathSciNet  Google Scholar 

  13. N. Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann., 195 (1972), 149–158.

    Article  MathSciNet  Google Scholar 

  14. F. Klein, Ueber den Verlauf der Abel’schen Integrale bei den Curven vierten Grades, Math. Ann., 10 (1876), 365–397.

    Article  MathSciNet  Google Scholar 

  15. B. Macdonald, Density of complex zeros of a system of real random polynomials, J. Stat. Phys., 136 (2009), 807–833.

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Nazarov and M. Sodin, On the number of nodal domains of random spherical harmonics, Am. J. Math., 131 (2009), 1337–1357.

    Article  MathSciNet  MATH  Google Scholar 

  17. B. Shiffman and S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Commun. Math. Phys., 200 (1999), 661–683.

    Article  MathSciNet  MATH  Google Scholar 

  18. B. Shiffman and S. Zelditch, Number variance of random zeros, Geom. Funct. Anal., 18 (2008), 1422–1475.

    Article  MathSciNet  MATH  Google Scholar 

  19. B. Shiffman, S. Zelditch, and S. Zrebiec, Overcrowding and hole probabilities for random zeros on complex manifolds, Indiana Univ. Math. J., 57 (2008), 1977–1997.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Sodin and B. Tsirelson, Random complex zeroes. I. Asymptotic normality, Isr. J. Math., 144 (2004), 125–149.

    Article  MathSciNet  MATH  Google Scholar 

  21. G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differ. Geom., 32 (1990), 99–130.

    MATH  Google Scholar 

  22. G. Wilson, Hilbert’s sixteenth problem, Topology, 17 (1978), 53–73.

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Zelditch, Szegö kernels and a theorem of Tian, Int. Math. Res. Not., 1998 (1998), 317–331.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, Université de Lyon, 69622, Villeurbanne Cedex, France

    Damien Gayet & Jean-Yves Welschinger

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  1. Damien Gayet
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  2. Jean-Yves Welschinger
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Correspondence to Jean-Yves Welschinger.

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Gayet, D., Welschinger, JY. Exponential rarefaction of real curves with many components. Publ.math.IHES 113, 69–96 (2011). https://doi.org/10.1007/s10240-011-0033-3

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  • DOI: https://doi.org/10.1007/s10240-011-0033-3

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