[Nombre et distribution attendue des points critiques de pinceaux de Lefschetz réels]
Dans cet article, on donne une formule de Riemann–Hurwitz asymptotique et probabiliste qui calcule la valeur attendue de l’indice de ramification réel d’un revêtement aléatoire de la sphère de Riemann. Plus généralement, on étudie l’asymptotique de la valeur attendue du nombre et de la distribution des points critiques réels d’un pinceau de Lefschetz réel sur une variété algébrique réelle. Tout au long de l’article, on donne des résultats analogues pour le cas complexe. Notre outil principal est la théorie des sections pics d’Hörmander.
We give an asymptotic probabilistic real Riemann–Hurwitz formula computing the expected real ramification index of a random covering over the Riemann sphere. More generally, we study the asymptotic expected number and distribution of critical points of a random real Lefschetz pencil over a smooth real algebraic variety. Throughout the paper, we give similar results for the complex case. Our main tool is Hörmander theory of peak sections.
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Keywords: real algebraic varieties, Lefschetz pencils, peak sections, random geometry
Mot clés : variétés algébriques réelles, pinceaux de Lefschetz, sections pics, géométrie aléatoire
@article{AIF_2020__70_3_1085_0,
author = {Ancona, Michele},
title = {Expected number and distribution of critical points of real {Lefschetz} pencils},
journal = {Annales de l'Institut Fourier},
pages = {1085--1113},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {70},
number = {3},
year = {2020},
doi = {10.5802/aif.3331},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.3331/}
}
TY - JOUR AU - Ancona, Michele TI - Expected number and distribution of critical points of real Lefschetz pencils JO - Annales de l'Institut Fourier PY - 2020 SP - 1085 EP - 1113 VL - 70 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3331/ DO - 10.5802/aif.3331 LA - en ID - AIF_2020__70_3_1085_0 ER -
%0 Journal Article %A Ancona, Michele %T Expected number and distribution of critical points of real Lefschetz pencils %J Annales de l'Institut Fourier %D 2020 %P 1085-1113 %V 70 %N 3 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3331/ %R 10.5802/aif.3331 %G en %F AIF_2020__70_3_1085_0
Ancona, Michele. Expected number and distribution of critical points of real Lefschetz pencils. Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1085-1113. doi : 10.5802/aif.3331. http://www.numdam.org/articles/10.5802/aif.3331/
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