46 research outputs found
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 are almost surely asymptotic to the
equilibrium measure of .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 -space of polynomials
endowed with varying measures of the form under
suitable assumptions on the weight functions .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
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