no. G7
Article de recherche - Analyse et géométrie complexes, Géométrie algébrique
An Exponential Rarefaction Result for Sub-Gaussian Real Algebraic Maximal Curves
[Un résultat de raréfaction exponentielle pour les courbes algébriques maximales réelles sous-Gaussiennes]
Bayraktar, Turgay1  ; Karaca, Emel2
1Faculty of Engineering and Natural Sciences, Sabancı University, İstanbul, 34956 Turkey
2Polatlı Faculty of Science and Arts, Ankara Hacı Bayram Veli University, Ankara, 06900 Turkey
Comptes Rendus. Mathématique, Tome 362 (2024) no. G7, pp. 779-788

We prove that maximal real algebraic curves associated with sub-Gaussian random real holomorphic sections of a smoothly curved ample line bundle are exponentially rare. This generalizes the result of Gayet and Welschinger [13] proved in the Gaussian case for positively curved real holomorphic line bundles.

Nous démontrons que les courbes algébriques réelles maximales associées aux sections holomorphes réelles sous-Gaussiennes d’un faisceau de lignes amples à courbure lisse sont exponentiellement raréfiées. Cela généralise le résultat de Gayet et Welschinger [13] ont prouvé dans le cas Gaussien des faisceaux de lignes holomorphes réels à courbure positive.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.596

Bayraktar, Turgay  1   ; Karaca, Emel  2

1 Faculty of Engineering and Natural Sciences, Sabancı University, İstanbul, 34956 Turkey
2 Polatlı Faculty of Science and Arts, Ankara Hacı Bayram Veli University, Ankara, 06900 Turkey
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{CRMATH_2024__362_G7_779_0,
     author = {Bayraktar, Turgay and Karaca, Emel},
     title = {An {Exponential} {Rarefaction} {Result} for {Sub-Gaussian} {Real} {Algebraic} {Maximal} {Curves}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {779--788},
     year = {2024},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     number = {G7},
     doi = {10.5802/crmath.596},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/crmath.596/}
}
TY  - JOUR
AU  - Bayraktar, Turgay
AU  - Karaca, Emel
TI  - An Exponential Rarefaction Result for Sub-Gaussian Real Algebraic Maximal Curves
JO  - Comptes Rendus. Mathématique
PY  - 2024
SP  - 779
EP  - 788
VL  - 362
IS  - G7
PB  - Académie des sciences, Paris
UR  - https://www.numdam.org/articles/10.5802/crmath.596/
DO  - 10.5802/crmath.596
LA  - en
ID  - CRMATH_2024__362_G7_779_0
ER  - 
%0 Journal Article
%A Bayraktar, Turgay
%A Karaca, Emel
%T An Exponential Rarefaction Result for Sub-Gaussian Real Algebraic Maximal Curves
%J Comptes Rendus. Mathématique
%D 2024
%P 779-788
%V 362
%N G7
%I Académie des sciences, Paris
%U https://www.numdam.org/articles/10.5802/crmath.596/
%R 10.5802/crmath.596
%G en
%F CRMATH_2024__362_G7_779_0
Bayraktar, Turgay; Karaca, Emel. An Exponential Rarefaction Result for Sub-Gaussian Real Algebraic Maximal Curves. Comptes Rendus. Mathématique, Tome 362 (2024) no. G7, pp. 779-788. doi: 10.5802/crmath.596

[1] Ancona, M. Exponential rarefaction of maximal real algebraic hypersurfaces (2020) (2009.11951v1)

[2] Bayraktar, T. Expected number of real roots for random linear combinations of orthogonal polynomials associated with radial weights, Potential Anal., Volume 48 (2018), pp. 459-471 | DOI | Zbl | MR

[3] Bayraktar, T. Mass equidistribution for random polynomials, Potential Anal., Volume 53 (2020), pp. 1403-1421 | DOI | Zbl | MR

[4] Berman, R. J. Bergman kernels and equilibrium measures for line bundles over projective manifolds, Am. J. Math., Volume 131 (2009) no. 5, pp. 1485-1524 | DOI | Zbl | MR

[5] Bedford, E.; Lyubich, M.; Smillie, J. Polynomial diffeomorphisms of C2. IV. The measure of maximal entropy and laminar currents, Invent. Math., Volume 112 (1993), pp. 77-125 | DOI | Zbl | MR

[6] Demailly, J.-P. Complex analytic and differential geometry (https://www.ime.usp.br/~cordaro/wp-content/uploads/2022/03/Jean-Pierre-Demailly-Complex-Analytic-and-Differential-Geometry-1.pdf)

[7] Demailly, J.-P. Courants positifs extrêmaux et conjecture de Hodge, Invent. Math., Volume 69 (1982) no. 3, pp. 347-374 | DOI | Zbl | MR

[8] Demailly, J.-P. Singular Hermitian metrics on positive line bundles, Complex algebraic varieties (Bayreuth, 1990) (Lecture Notes in Mathematics), Volume 1507, Springer, 1990, pp. 87-104 | Zbl | DOI | MR

[9] Diatta, D. N.; Lerario, A. Low-degree approximation of random polynomials, Found. Comput. Math., Volume 22 (2022), pp. 77-97 | DOI | Zbl | MR

[10] Dujardin, R. Laminar currents in P 2 , Math. Ann., Volume 325 (2003), pp. 745-765 | DOI | Zbl | MR

[11] Dujardin, R. Structure properties of laminar currents on P 2 , J. Geom. Anal., Volume 15 (2005) no. 1, pp. 25-47 | DOI | Zbl | MR

[12] Fornæss, J. E.; Sibony, N. Oka’s inequality for currents and applications, Math. Ann., Volume 301 (1995) no. 3, pp. 399-419 | DOI | Zbl | MR

[13] Griffiths, P.; Harris, J. Principles of algebraic geometry, Pure and Applied Mathematics, John Wiley & Sons, 1978 | Zbl | MR

[14] Gayet, D.; Welschinger, J.-Y. Exponential rarefaction of real curves with many components, Publ. Math., Inst. Hautes Étud. Sci., Volume 113 (2011), pp. 69-96 | DOI | Zbl | Numdam | MR

[15] Hanson, D. L.; Wright, F. T. A bound on tail probabilities for quadratic forms in independent random variables, Ann. Math. Stat., Volume 42 (1971), pp. 1079-1083 | DOI | Zbl | MR

[16] Rudelson, M.; Vershynin, R. Hanson–Wright inequality and sub-Gaussian concentration, Electron. Commun. Probab., Volume 82 (2013) no. 9, 82 | DOI | Zbl | MR

[17] Shiffman, B.; Zelditch, S. Distribution of zeros of random and quantum chaotic sections of positive line bundles, Commun. Math. Phys., Volume 200 (1999) no. 3, pp. 661-683 | DOI | Zbl | MR

[18] Vershynin, R. Introduction to the non-asymptotic analysis of random matrices, Compressed sensing: Theory and Applications, Cambridge University Press, 2012, pp. 210-268 | DOI

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

[20] Zelditch, S. Quantum ergodic sequences and equilibrium measures, Constr. Approx., Volume 47 (2018) no. 1, pp. 89-118 | DOI | Zbl | MR

Cité par Sources :