Equidistribution of zeros of random holomorphic sections

We study asymptotic distribution of zeros of random holomorphic sections of high powers of positive line bundles defined over projective homogenous manifolds. We work with a wide class of distributions that includes real and complex Gaussians. As a special case, we obtain asymptotic zero distribution of multivariate complex polynomials given by linear combinations of orthogonal polynomials with i.i.d. random coefficients. Namely, we prove that normalized zero measures of m i.i.d random polynomials, orthonormalized on a regular compact set $K\subset \Bbb{C}^m,$ are almost surely asymptotic to the equilibrium measure of $K$.Comment: Final version incorporates referee comments. To appear in Indiana Univ. Math.

Asymptotic normality of linear statistics of zeros of random polynomials

In this note, we prove a central limit theorem for smooth linear statistics of zeros of random polynomials which are linear combinations of orthogonal polynomials with iid standard complex Gaussian coefficients. Along the way, we obtain Bergman kernel asymptotics for weighted $L^2$-space of polynomials endowed with varying measures of the form $e^{-2n\varphi_n(z)}dz$ under suitable assumptions on the weight functions $\varphi_n$.Comment: Minor revisions, references added. To appear in Proc. of Amer. Math. So

Constraints on automorphism groups of higher dimensional manifolds

In this note, we prove, for instance, that the automorphism group of a rational manifold X which is obtained from CP^k by a finite sequence of blow-ups along smooth centers of dimension at most r with k>2r+2 has finite image in GL(H^*(X,Z)). In particular, every holomorphic automorphism $f:X\to X$ has zero topological entropy

On Dynamics of Asymptotically Minimal Polynomials

We study dynamical properties of asymptotically extremal polynomials associated with a non-polar planar compact set E. In particular, we prove that if the zeros of such polynomials are uniformly bounded then their Brolin measures converge weakly to the equilibrium measure of E. In addition, if E is regular and the zeros of such polynomials are sufficiently close to E then we prove that the filled Julia sets converge to polynomial convex hull of E in the Klimek topology

Zero Distribution of Random Bernoulli Polynomial Mappings

In this note, we study asymptotic zero distribution of multivariable full system of random polynomials with independent Bernoulli coefficients. We prove that with overwhelming probability their simultaneous zeros sets are discrete and the associated normalized empirical measure of zeros asymptotic to the Haar measure on the unit torus.Comment: Minor revisions. To appear in Electron. J. Proba

An Exponential Rarefaction Result for Sub-Gaussian Real Algebraic Maximal Curves

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 \cite{GW} proved in the Gaussian case for positively curved real holomorphic line bundles.Comment: minor revision