Expected number and distribution of critical points of real Lefschetz pencils
Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1085-1113.

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.

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.

Received: 2017-07-26
Revised: 2019-04-15
Accepted: 2019-09-18
Published online: 2020-06-26
DOI: https://doi.org/10.5802/aif.3331
Classification: 14P99,  32U40,  60D05
Keywords: real algebraic varieties, Lefschetz pencils, peak sections, random geometry
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {3},
     year = {2020},
     pages = {1085-1113},
     doi = {10.5802/aif.3331},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_3_1085_0/}
}
Ancona, Michele. Expected number and distribution of critical points of real Lefschetz pencils. Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1085-1113. doi : 10.5802/aif.3331. https://aif.centre-mersenne.org/item/AIF_2020__70_3_1085_0/

[1] Bürgisser, Peter Average Euler characteristic of random real algebraic varieties, C. R. Math. Acad. Sci. Paris, Volume 345 (2007) no. 9, pp. 507-512 | Article | MR 2375112 | Zbl 1141.60033

[2] Federer, Herbert Geometric measure theory, Grundlehren der Mathematischen Wissenschaften, Volume 153, Springer, 1969, xiv+676 pages | MR 0257325 | Zbl 0176.00801

[3] Gayet, Damien; Welschinger, Jean-Yves What is the total Betti number of a random real hypersurface?, J. Reine Angew. Math., Volume 689 (2014), pp. 137-168 | Article | MR 3187930 | Zbl 1348.14138

[4] Gayet, Damien; Welschinger, Jean-Yves Expected topology of random real algebraic submanifolds, J. Inst. Math. Jussieu, Volume 14 (2015) no. 4, pp. 673-702 | Article | MR 3394124 | Zbl 1326.32040

[5] Gayet, Damien; Welschinger, Jean-Yves Betti numbers of random real hypersurfaces and determinants of random symmetric matrices, J. Eur. Math. Soc., Volume 18 (2016) no. 4, pp. 733-772 | Article | MR 3474455 | Zbl 1408.14187

[6] Hörmander, Lars An introduction to complex analysis in several variables, North-Holland Mathematical Library, Volume 7, North-Holland, 1990, xii+254 pages | MR 1045639 | Zbl 0685.32001

[7] Kac, Mark A correction to “On the average number of real roots of a random algebraic equation”, Bull. Am. Math. Soc., Volume 49 (1943), 938 pages | Article | MR 0009655 | Zbl 0060.28603

[8] Kostlan, Eric On the distribution of roots of random polynomials, From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), Springer, 1993, pp. 419-431 | Article | MR 1246137 | Zbl 0788.60069

[9] Lerario, Antonio; Lundberg, Erik Statistics on Hilbert’s 16th problem, Int. Math. Res. Not. (2015) no. 12, pp. 4293-4321 | Article | MR 3356754 | Zbl 1396.14049

[10] Lerario, Antonio; Lundberg, Erik Gap probabilities and Betti numbers of a random intersection of quadrics, Discrete Comput. Geom., Volume 55 (2016) no. 2, pp. 462-496 | Article | MR 3458605 | Zbl 1358.14040

[11] Letendre, Thomas Expected volume and Euler characteristic of random submanifolds, J. Funct. Anal., Volume 270 (2016) no. 8, pp. 3047-3110 | Article | MR 3470435 | Zbl 1349.58007

[12] Lundberg, Erik; Ramachandran, Koushik The arc length and topology of a random lemniscate, J. Lond. Math. Soc., II. Ser., Volume 96 (2017) no. 3, pp. 621-641 | Article | MR 3742436 | Zbl 1417.60040

[13] Nicolaescu, Liviu I. Critical sets of random smooth functions on compact manifolds, Asian J. Math., Volume 19 (2015) no. 3, pp. 391-432 | Article | MR 3361277 | Zbl 1341.60081

[14] Podkorytov, Semën S. On the Euler characteristic of a random algebraic hypersurface, Zap. Nauchn. Semin. (POMI), Volume 252 (1998), pp. 224-230 | Article | MR 1756726 | Zbl 1021.14017

[15] Shiffman, Bernard; Zelditch, Steve Distribution of zeros of random and quantum chaotic sections of positive line bundles, Commun. Math. Phys., Volume 200 (1999) no. 3, pp. 661-683 | Article | MR 1675133 | Zbl 0919.32020

[16] Shub, Michael; Smale, Steve Complexity of Bezout’s theorem. II. Volumes and probabilities, Computational algebraic geometry (Nice, 1992) (Progress in Mathematics) Volume 109, Birkhäuser, 1993, pp. 267-285 | Article | MR 1230872 | Zbl 0851.65031

[17] Tian, Gang On a set of polarized Kähler metrics on algebraic manifolds, J. Differ. Geom., Volume 32 (1990) no. 1, pp. 99-130 | Article | MR 1064867 | Zbl 0706.53036