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.
@article{PMIHES_2011__113__69_0, author = {Gayet, Damien and Welschinger, Jean-Yves}, title = {Exponential rarefaction of real curves with many components}, journal = {Publications Math\'ematiques de l'IH\'ES}, pages = {69--96}, publisher = {Springer-Verlag}, volume = {113}, year = {2011}, doi = {10.1007/s10240-011-0033-3}, mrnumber = {2805598}, zbl = {1227.32028}, language = {en}, url = {https://www.numdam.org/articles/10.1007/s10240-011-0033-3/} }
TY - JOUR AU - Gayet, Damien AU - Welschinger, Jean-Yves TI - Exponential rarefaction of real curves with many components JO - Publications Mathématiques de l'IHÉS PY - 2011 SP - 69 EP - 96 VL - 113 PB - Springer-Verlag UR - https://www.numdam.org/articles/10.1007/s10240-011-0033-3/ DO - 10.1007/s10240-011-0033-3 LA - en ID - PMIHES_2011__113__69_0 ER -
%0 Journal Article %A Gayet, Damien %A Welschinger, Jean-Yves %T Exponential rarefaction of real curves with many components %J Publications Mathématiques de l'IHÉS %D 2011 %P 69-96 %V 113 %I Springer-Verlag %U https://www.numdam.org/articles/10.1007/s10240-011-0033-3/ %R 10.1007/s10240-011-0033-3 %G en %F PMIHES_2011__113__69_0
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. https://www.numdam.org/articles/10.1007/s10240-011-0033-3/
[1.] 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.] A direct approach to Bergman kernel asymptotics for positive line bundles, Ark. Mat., Volume 46 (2008), pp. 1871-2487 | DOI | MR | Zbl
[3.] 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.] 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.] Sur la laminarité de certains courants, Ann. Sci. Ecole Norm. Super., Volume 37 (2004), pp. 304-311 | Numdam | MR | Zbl
[6.] Courants positifs extrêmaux et conjecture de Hodge, Invent. Math., Volume 69 (1982), pp. 347-374 | DOI | MR | Zbl
[7.] Distribution des valeurs de transformations méromorphes et applications, Comment. Math. Helv., Volume 81 (2006), pp. 221-258 | DOI | MR | Zbl
[8.] Symplectic submanifolds and almost-complex geometry, J. Differ. Geom., Volume 44 (1996), pp. 666-705 | MR | Zbl
[9.] Laminar currents in ${\Bbb{P}}^{2}$ , Math. Ann., Volume 325 (2003), pp. 745-765 | DOI | MR | Zbl
[10.] How many zeros of a random polynomial are real?, Bull. Am. Math. Soc., Volume 32 (1995), pp. 1-37 | DOI | MR | Zbl
[11.] Symplectic real hypersurfaces and Lefschetz pencils, J. Symplectic Geom., Volume 6 (2008), pp. 247-266 | MR | Zbl
[12.] Ueber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann., Volume 10 (1876), pp. 189-198 | DOI | JFM | MR
[13.] The Bergman kernel function. Differentiability at the boundary, Math. Ann., Volume 195 (1972), pp. 149-158 | DOI | MR | Zbl
[14.] Ueber den Verlauf der Abel’schen Integrale bei den Curven vierten Grades, Math. Ann., Volume 10 (1876), pp. 365-397 | DOI | JFM | MR
[15.] Density of complex zeros of a system of real random polynomials, J. Stat. Phys., Volume 136 (2009), pp. 807-833 | DOI | MR | Zbl
[16.] On the number of nodal domains of random spherical harmonics, Am. J. Math., Volume 131 (2009), pp. 1337-1357 | DOI | MR | Zbl
[17.] 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.] Number variance of random zeros, Geom. Funct. Anal., Volume 18 (2008), pp. 1422-1475 | DOI | MR | Zbl
[19.] Overcrowding and hole probabilities for random zeros on complex manifolds, Indiana Univ. Math. J., Volume 57 (2008), pp. 1977-1997 | DOI | MR | Zbl
[20.] Random complex zeroes. I. Asymptotic normality, Isr. J. Math., Volume 144 (2004), pp. 125-149 | DOI | MR | Zbl
[21.] On a set of polarized Kähler metrics on algebraic manifolds, J. Differ. Geom., Volume 32 (1990), pp. 99-130 | MR | Zbl
[22.] Hilbert’s sixteenth problem, Topology, Volume 17 (1978), pp. 53-73 | DOI | MR | Zbl
[23.] Szegö kernels and a theorem of Tian, Int. Math. Res. Not., Volume 1998 (1998), pp. 317-331 | DOI | MR | Zbl
- Exponential concentration for the number of roots of random trigonometric polynomials, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 60 (2024) no. 2 | DOI:10.1214/23-aihp1366
- An Exponential Rarefaction Result for Sub-Gaussian Real Algebraic Maximal Curves, Comptes Rendus. Mathématique, Volume 362 (2024) no. G7, p. 779 | DOI:10.5802/crmath.596
- Upper estimates for the expected Betti numbers of random subcomplexes, Topology and its Applications, Volume 355 (2024), p. 109010 | DOI:10.1016/j.topol.2024.109010
- On the Topology of Random Real Complete Intersections, The Journal of Geometric Analysis, Volume 33 (2023) no. 1 | DOI:10.1007/s12220-022-01092-x
- Limit cycle enumeration in random vector fields, Transactions of the American Mathematical Society, Volume 376 (2023) no. 8, p. 5693 | DOI:10.1090/tran/8936
- Low-Degree Approximation of Random Polynomials, Foundations of Computational Mathematics, Volume 22 (2022) no. 1, p. 77 | DOI:10.1007/s10208-021-09506-y
- Asymptotic topology of excursion and nodal sets of Gaussian random fields, Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2022 (2022) no. 790, p. 149 | DOI:10.1515/crelle-2022-0027
- RANDOM REAL BRANCHED COVERINGS OF THE PROJECTIVE LINE, Journal of the Institute of Mathematics of Jussieu, Volume 21 (2022) no. 5, p. 1783 | DOI:10.1017/s1474748020000742
- Mean conservation of nodal volume and connectivity measures for Gaussian ensembles, Advances in Mathematics, Volume 378 (2021), p. 107521 | DOI:10.1016/j.aim.2020.107521
- Roots of Kostlan polynomials: moments, strong Law of Large Numbers and Central Limit Theorem, Annales Henri Lebesgue, Volume 4 (2021), p. 1659 | DOI:10.5802/ahl.113
- Zeros of smooth stationary Gaussian processes, Electronic Journal of Probability, Volume 26 (2021) no. none | DOI:10.1214/21-ejp637
- Random Sections of Line Bundles Over Real Riemann Surfaces, International Mathematics Research Notices, Volume 2021 (2021) no. 9, p. 7004 | DOI:10.1093/imrn/rnz051
- Quasi-independence for nodal lines, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 55 (2019) no. 3 | DOI:10.1214/18-aihp931
- The arc length and topology of a random lemniscate, Journal of the London Mathematical Society, Volume 96 (2017) no. 3, p. 621 | DOI:10.1112/jlms.12086
- Percolation of random nodal lines, Publications mathématiques de l'IHÉS, Volume 126 (2017) no. 1, p. 131 | DOI:10.1007/s10240-017-0093-0
- Two-Point Correlation Functions and Universality for the Zeros of Systems of SO(n+1)-invariant Gaussian Random Polynomials, International Mathematics Research Notices, Volume 2016 (2016) no. 11, p. 3237 | DOI:10.1093/imrn/rnv236
- EXPECTED TOPOLOGY OF RANDOM REAL ALGEBRAIC SUBMANIFOLDS, Journal of the Institute of Mathematics of Jussieu, Volume 14 (2015) no. 4, p. 673 | DOI:10.1017/s1474748014000115
- Statistics on Hilbert's 16th Problem, International Mathematics Research Notices (2014) | DOI:10.1093/imrn/rnu069
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