Let L be a positive line bundle over a Riemann surface Σ defined over R. We prove that sections s of L d , d ≫ 0, whose number of real zeros #Zs deviates from the expected one are rare. We also provide asymptotics of the form

Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverte www.centre-mersenne.org Michele A Expected number and distribution of critical points of real Lefschetz pencils Tome , n o (), p.- .

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We study the number of real roots of a Kostlan (or elliptic) random polynomial of degree d in one variable. More generally, we are interested in the distribution of the counting measure of the set of real roots of such a polynomial. We compute the asymptotics of the central moments of any order of these random variables, in the large degree limit. As a consequence, we prove that these quantities satisfy a strong Law of Large Numbers and a Central Limit Theorem. In particular, the real roots of a Kostlan polynomial almost surely equidistribute as the degree diverges. Moreover, the fluctuations of the counting measure of this random set around its mean converge in distribution to the Gaussian White Noise. We also prove that the random variables we study concentrate in probability around their mean faster than any negative power of d. More generally, our results hold for the real zeros of a random real section of a line bundle of degree d over a real projective curve, in the complex Fubini-Study model.