Steve Zelditch

Szegö kernels and a theorem of Tian

We give a simple proof of Tians theorem that the Kodaira embeddings associated to a positive line bundle over a compact complex manifold are asymptotically isometric. The proof is based on the diagonal asymptotics of the Szego kernel (i.e. the orthogonal projection onto holomorphic sections). In deriving these asymptotics we use the Boutet de Monvel-Sjostrand parametrix for the Szego kernel.

Distribution of Zeros of Random and Quantum Chaotic Sections of Positive Line Bundles

Abstract:We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers MN of a positive holomorphic Hermitian line bundle L over a compact complex manifold M. Our first result concerns “random” sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases {SNj} of H0(M, LN), we show that for almost every sequence {SNj}, the associated sequence of zero currents &1/NZSNj; tends to the curvature form ω of L. Thus, the zeros of a sequence of sections sN∈H0(M, LN) chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases {SNj} of H0(M, LN) consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.

Universality and scaling of correlations between zeros on complex manifolds

Abstract.We study the limit as N→∞ of the correlations between simultaneous zeros of random sections of the powers LN of a positive holomorphic line bundle L over a compact complex manifold M, when distances are rescaled so that the average density of zeros is independent of N. We show that the limit correlation is independent of the line bundle and depends only on the dimension of M and the codimension of the zero sets. We also provide some explicit formulas for pair correlations. In particular, we prove that Hannay’s limit pair correlation function for SU(2) polynomials holds for all compact Riemann surfaces.

Riemannian manifolds with maximal eigenfunction growth

On any compact Riemannian manifold

Critical points and supersymmetric vacua I

(M, g)

Spectral determination of analytic bi-axisymmetric plane domains

of dimension

EQUILIBRIUM DISTRIBUTION OF ZEROS OF RANDOM POLYNOMIALS

n

Riemannian manifolds with uniformly bounded eigenfunctions

, the

Poincaré-Lelong Approach to Universality and Scaling of Correlations Between Zeros

L^2

Critical Points and Supersymmetric Vacua, III: String/M Models

-normalized eigenfunctions