Bernard Shiffman

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.

A Characterization of Holomorphic Chains

It was observed by P. Lelong [16] that a pure k-dimensional analytic subvariety V of an open set Q contained in Cm determines a current [ V] on Q via integration over the regular points of V. The current [ V] has several special properties: It is of type (m k, m k), positive, d-closed, and locally rectifiable. Conversely, J. King [14] proved that every current with these properties corresponds to integration over a pure k-dimensional variety with positive integral multiplicities. In this paper, we generalize Kings theorem by dropping the assumption that the current be positive and then concluding (with an added assumption on the size of the support) that the current corresponds to integration over a pure k-dimensional variety with arbitrary integral multiplicities (Theorem 2.1). Such currents, which represent analytic cycles of dimension k, are called holomorphic k-chains (Definition 1.7) and are characterized by Theorem 2.1. Our techniques involve Federers geometric theory of currents [7]. In Section 1.1 we describe the locally rectifiable currents of Federer and Fleming [8], for which we provide a new characterization (Theorems 1.4 and 1.5). Using Theorems 1.5 and 2.1, we can also characterize the holomorphic k-chains as the d-closed currents of type (m k, m k) that are representable by integration and have positive integral 2k-densities almost everywhere (Theorem 2.2). Our characterization Theorem 2.1 can be used to prove that all stable, d-closed, locally rectifiable currents in CPm (complex projective m-space) are algebraic cycles. This application, which inspired this paper, is due to Lawson and Simons [15]. We also prove that if a complex subvariety is a solution to a Plateau problem in Cm, then it is the only solution (Theorem 3.6). As a special case of Theorem 2.1 we obtain a new, somewhat simpler proof of Kings theorem characterizing the positive holomorphic k-chains. (Another proof and generalization of Kings theorem is given in [11], which uses a result of E. Bombieri [3], [4].) We also use our characterization theorem to deduce a theorem of E. Bishop [1] on extending analytic subvarieties with finite volume.

Critical points and supersymmetric vacua I

Supersymmetric vacua (‘universes’) of string/M theory may be identified with certain critical points of a holomorphic section (the ‘superpotential’) of a Hermitian holomorphic line bundle over a complex manifold. An important physical problem is to determine how many vacua there are and how they are distributed, as the superpotential varies over physically relevant ensembles. In several papers over the last few years, M. R. Douglas and co-workers have studied such vacuum statistics problems for a variety of physical models at the physics level of rigor [Do,AD,DD]. The present paper is the first of a series by the present authors giving a rigorous mathematical foundation for the vacuum statistics problem. It sets down basic results on the statistics of critical points ∇s=0 of random holomorphic sections of Hermitian holomorphic line bundles with respect to a metric connection ∇, when the sections are endowed with a Gaussian measure. The principal results give formulas for the expected density and number of critical points of fixed Morse index of Gaussian random sections relative to ∇. They are particularly concrete for Riemann surfaces. In our subsequent work, the results will be applied to the vacuum statistics problem and to the purely geometric problem of studying metrics which minimize the expected number of critical points.

EQUILIBRIUM DISTRIBUTION OF ZEROS OF RANDOM POLYNOMIALS

We consider ensembles of random polynomials of the form p(z) = P N=1 ajPj where {aj} are independent complex normal random variables and where {Pj} are the orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain ⊂ C relative to an analytic weight �(z)|dz| . In the simplest case where is the unit disk and � = 1, so that Pj(z) = z j , it is known that the average distribution of zeros is the uniform measure on S 1 . We show that for any analytic (,�), the zeros of random polynomials almost surely become equidistributed relative to the equilibrium measure on @ as N → ∞. We further show that on the length scale of 1/N, the correlations have a universal scaling limit independent of (,�).

Properties of compact complex manifolds carrying closed positive currents

We show that a compact complex manifold is Moishezon if and only if it carries a strictly positive, integral (1, 1)-current. We then study holomorphic line bundles carrying singular hermitian metrics with semi-positive curvature currents, and we give some cases in which these line bundles are big. We use these cases to provide sufficient conditions for a compact complex manifold to be Moishezon in terms of the existence of certain semi-positive, integral (1,1)-currents. We also show that the intersection number of two closed semi-positive currents of complementary degrees on a compact complex manifold is positive when the intersection of their singular supports is contained in a Stein domain.

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

Abstract:This note is concerned with the scaling limit as N→∞ of n-point correlations between zeros of random holomorphic polynomials of degree N in m variables. More generally we study correlations between zeros of holomorphic sections of powers LN of any positive holomorphic line bundle L over a compact Kähler manifold. Distances are rescaled so that the average density of zeros is independent of N. Our main result is that the scaling limits of the correlation functions and, more generally, of the “correlation forms” are universal, i.e. independent of the bundle L, manifold M or point on M.

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

AbstractA fundamental problem in contemporary string/M theory is to count the number of inequivalent vacua satisfying constraints in a string theory model. This article contains the first rigorous results on the number and distribution of supersymmetric vacua of type IIb string theories compactified on a Calabi-Yau 3-fold X with flux. In particular, complete proofs of the counting formulas in Ashok-Douglas [AD] and Denef-Douglas [DD1] are given, together with van der Corput style remainder estimates.Supersymmetric vacua are critical points of certain holomorphic sections (flux superpotentials) of a line bundle

Uniqueness of entire and meromorphic functions sharing finite sets

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