Critical values of random analytic functions on complex manifolds

We study the asymptotic distribution of critical values of random holomorphic sections sn ∈ H0(Mm, Ln) of powers of a positive line bundle (L,h) → (M,) on a general Kähler manifold of dimension m. By critical value is meant the value of |s(z)|hn at a critical point where ▿hsn(z) = 0, where ▿h is the Chern connection. The distribution of critical values of sn is its empirical measure. Two main ensembles are considered: (i) the normalized Gaussian ensembles so that Eksnk2 L2 = 1; (ii) the spherical ensemble defined by Haar measure on the unit sphere SH0(M, Ln) ▿ H0(M, Ln) with ksnk2 L2 = 1.

The main result is that the expected distributions of critical values in both the normalized Gaussian ensemble and the spherical ensemble tend to the same universal limit as n → ∞, given explicitly as an integral over m×m symmetric matrices.