Exponential rarefaction of real curves with many components
Gayet, Damien1 ; Welschinger, Jean-Yves1
1 CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, Université de Lyon 69622, Villeurbanne Cedex France
Publications Mathématiques de l'IHÉS, Tome 113 (2011), pp. 69-96.

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.

DOI : 10.1007/s10240-011-0033-3
Gayet, Damien 1 ; Welschinger, Jean-Yves 1

1 CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, Université de Lyon 69622, Villeurbanne Cedex France 
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Gayet, Damien; Welschinger, Jean-Yves. Exponential rarefaction of real curves with many components. Publications Mathématiques de l'IHÉS, Tome 113 (2011), pp. 69-96. doi : 10.1007/s10240-011-0033-3. http://archive.numdam.org/articles/10.1007/s10240-011-0033-3/

[1.] Bedford, E.; Lyubich, M.; Smillie, J. Polynomial diffeomorphisms of C 2. IV. The measure of maximal entropy and laminar currents, Invent. Math., Volume 112 (1993), pp. 77-125 | DOI | MR | Zbl

[2.] Berman, R.; Berndtsson, B.; Sjöstrand, J. A direct approach to Bergman kernel asymptotics for positive line bundles, Ark. Mat., Volume 46 (2008), pp. 1871-2487 | DOI | MR | Zbl

[3.] Bouche, T. Convergence de la métrique de Fubini-Study d’un fibré linéaire positif, Ann. Inst. Fourier, Volume 40 (1990), pp. 117-130 | Numdam | MR | Zbl

[4.] Catlin, E. The Bergman Kernel and a theorem of Tian, Analysis and Geometry in Several Complex Variables, Birkhäuser, Boston, 1999 (G. Komatsu and M. Kuranishi, eds) | MR | Zbl

[5.] Thélin, H. Sur la laminarité de certains courants, Ann. Sci. Ecole Norm. Super., Volume 37 (2004), pp. 304-311 | Numdam | MR | Zbl

[6.] Demailly, J.-P. Courants positifs extrêmaux et conjecture de Hodge, Invent. Math., Volume 69 (1982), pp. 347-374 | DOI | MR | Zbl

[7.] Dinh, T.-C.; Sibony, N. Distribution des valeurs de transformations méromorphes et applications, Comment. Math. Helv., Volume 81 (2006), pp. 221-258 | DOI | MR | Zbl

[8.] Donaldson, S. K. Symplectic submanifolds and almost-complex geometry, J. Differ. Geom., Volume 44 (1996), pp. 666-705 | MR | Zbl

[9.] Dujardin, R. Laminar currents in ${\Bbb{P}}^{2}$ , Math. Ann., Volume 325 (2003), pp. 745-765 | DOI | MR | Zbl

[10.] Edelman, A.; Kostlan, E. How many zeros of a random polynomial are real?, Bull. Am. Math. Soc., Volume 32 (1995), pp. 1-37 | DOI | MR | Zbl

[11.] Gayet, D. Symplectic real hypersurfaces and Lefschetz pencils, J. Symplectic Geom., Volume 6 (2008), pp. 247-266 | MR | Zbl

[12.] Harnack, A. Ueber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann., Volume 10 (1876), pp. 189-198 | DOI | JFM | MR

[13.] Kerzman, N. The Bergman kernel function. Differentiability at the boundary, Math. Ann., Volume 195 (1972), pp. 149-158 | DOI | MR | Zbl

[14.] Klein, F. Ueber den Verlauf der Abel’schen Integrale bei den Curven vierten Grades, Math. Ann., Volume 10 (1876), pp. 365-397 | DOI | JFM | MR

[15.] Macdonald, B. Density of complex zeros of a system of real random polynomials, J. Stat. Phys., Volume 136 (2009), pp. 807-833 | DOI | MR | Zbl

[16.] Nazarov, F.; Sodin, M. On the number of nodal domains of random spherical harmonics, Am. J. Math., Volume 131 (2009), pp. 1337-1357 | DOI | MR | Zbl

[17.] Shiffman, B.; Zelditch, S. Distribution of zeros of random and quantum chaotic sections of positive line bundles, Commun. Math. Phys., Volume 200 (1999), pp. 661-683 | DOI | MR | Zbl

[18.] Shiffman, B.; Zelditch, S. Number variance of random zeros, Geom. Funct. Anal., Volume 18 (2008), pp. 1422-1475 | DOI | MR | Zbl

[19.] Shiffman, B.; Zelditch, S.; Zrebiec, S. Overcrowding and hole probabilities for random zeros on complex manifolds, Indiana Univ. Math. J., Volume 57 (2008), pp. 1977-1997 | DOI | MR | Zbl

[20.] Sodin, M.; Tsirelson, B. Random complex zeroes. I. Asymptotic normality, Isr. J. Math., Volume 144 (2004), pp. 125-149 | DOI | MR | Zbl

[21.] Tian, G. On a set of polarized Kähler metrics on algebraic manifolds, J. Differ. Geom., Volume 32 (1990), pp. 99-130 | MR | Zbl

[22.] Wilson, G. Hilbert’s sixteenth problem, Topology, Volume 17 (1978), pp. 53-73 | DOI | MR | Zbl

[23.] Zelditch, S. Szegö kernels and a theorem of Tian, Int. Math. Res. Not., Volume 1998 (1998), pp. 317-331 | DOI | MR | Zbl

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