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Jun 30, 2018
06/18
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Bernard Shiffman
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We show the existence of uniformly bounded sequences of increasing numbers of orthonormal sections of powers $L^k$ of a positive holomorphic line bundle $L$ on a compact K\"ahler manifold $M$. In particular, we construct for each positive integer $k$, orthonormal sections $s^k_1,\dots,s^k_{n_k}$ in $H^0(M,L^k)$, $n_k\ge\beta\dim H^0(M,L^k)$, such that $\{s^k_j\}$ is a uniformly bounded family, where $\beta$ is an explicit positive constant depending only on the dimension of $M$. For $m=1$,...
Topics: Complex Variables, Mathematics, Differential Geometry, Algebraic Geometry
Source: http://arxiv.org/abs/1404.1508
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Sep 19, 2013
09/13
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Bernard Shiffman
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We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized distribution of zeros of systems of m polynomials of degree N, orthonormalized on a regular compact subset K of C^m, almost surely converge to the equilibrium measure on K as the degree N goes to infinity.
Source: http://arxiv.org/abs/0708.2754v1
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Sep 18, 2013
09/13
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Thomas Bloom; Bernard Shiffman
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For a regular compact set $K$ in $C^m$ and a measure $\mu$ on $K$ satisfying the Bernstein-Markov inequality, we consider the ensemble $P_N$ of polynomials of degree $N$, endowed with the Gaussian probability measure induced by $L^2(\mu)$. We show that for large $N$, the simultaneous zeros of $m$ polynomials in $P_N$ tend to concentrate around the Silov boundary of $K$; more precisely, their expected distribution is asymptotic to $N^m \mu_{eq}$, where $\mu_{eq}$ is the equilibrium measure of...
Source: http://arxiv.org/abs/math/0605739v1
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Sep 22, 2013
09/13
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Bernard Shiffman; Steve Zelditch
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The spaces $H^0(M, L^N)$ of holomorphic sections of the powers of an ample line bundle $L$ over a compact K\"ahler manifold $(M,\omega)$ have been generalized by Boutet de Monvel and Guillemin to spaces $H^0_J(M, L^N)$ of `almost holomorphic sections' of ample line bundles over an almost complex symplectic manifold $(M, J, \omega)$. We consider the unit spheres $SH^0_J(M, L^N)$ in the spaces $H^0_J(M, L^N)$, which we equip with natural inner products. Our purpose is to show that, in a...
Source: http://arxiv.org/abs/math/0001102v2
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Sep 20, 2013
09/13
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Bernard Shiffman; Steve Zelditch
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We show that the variance of the number of simultaneous zeros of $m$ i.i.d. Gaussian random polynomials of degree $N$ in an open set $U \subset C^m$ with smooth boundary is asymptotic to $N^{m-1/2} \nu_{mm} Vol(\partial U)$, where $\nu_{mm}$ is a universal constant depending only on the dimension $m$. We also give formulas for the variance of the volume of the set of simultaneous zeros in $U$ of $k
Source: http://arxiv.org/abs/math/0608743v3
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Sep 22, 2013
09/13
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Sergei Ivashkovich; Bernard Shiffman
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We prove that a meromorphic map defined on the complement of a compact subset of a three-dimensional Stein manifold M and with values in a compact complex three-fold X extends to the complement of a finite set of points. If X is simply connected, then the map extends to all of M.
Source: http://arxiv.org/abs/math/0003103v1
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Sep 18, 2013
09/13
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Bernard Shiffman; Mikhail Zaidenberg
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In this note we show that there are algebraic families of hyperbolic, Fermat-Waring type hypersurfaces in P^n of degree 4(n-1)^2, for all dimensions n>1. Moreover, there are hyperbolic Fermat-Waring hypersurfaces in P^n of degree 4n^2-2n+1 possessing complete hyperbolic, hyperbolically embedded complements.
Source: http://arxiv.org/abs/math/0101126v1
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Sep 17, 2013
09/13
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Bernard Shiffman; Steve Zelditch
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We consider the zero sets $Z_N$ of systems of $m$ random polynomials of degree $N$ in $m$ complex variables, and we give asymptotic formulas for the random variables given by summing a smooth test function over $Z_N$. Our asymptotic formulas show that the variances for these smooth statistics have the growth $N^{m-2}$. We also prove analogues for the integrals of smooth test forms over the subvarieties defined by $k
Source: http://arxiv.org/abs/0711.1840v1
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Sep 21, 2013
09/13
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Zhiqin Lu; Bernard Shiffman
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We compute the first four coefficients of the asymptotic off-diagonal expansion of the Bergman kernel for the N-th power of a positive line bundle on a compact Kaehler manifold, and we show that the coefficient b_1 of the N^{-1/2} term vanishes when we use a K-frame. We also show that all the coefficients of the expansion are polynomials in the K-coordinates and the covariant derivatives of the curvature and are homogeneous with respect to the weight w.
Source: http://arxiv.org/abs/1301.2166v2
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Sep 18, 2013
09/13
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Alexander Russakovskii; Bernard Shiffman
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We obtain results on the asymptotic equidistribution of the pre-images of linear subspaces for sequences of rational mappings between projective spaces. As an application to complex dynamics, we consider the iterates $P_k$ of a rational mapping $P$ of $\PP^n$. We show, assuming a condition on the topological degree $\lambda$ of $P$, that there is a probability measure $\mu$ on $\PP^n$ such that the discrete measures $\lambda^{-k}P_k^*\delta_w$ converge to $\mu$ for all $w\in\PP^n$ outside a...
Source: http://arxiv.org/abs/math/9604204v1
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Sep 18, 2013
09/13
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Bernard Shiffman; Mikhail Zaidenberg
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We describe a new method of constructing Kobayashi-hyperbolic surfaces in complex projective 3-space based on deforming surfaces with a "hyperbolic non-percolation" property. We use this method to show that general small deformations of certain singular abelian surfaces of degree 8 are hyperbolic. We also show that a union of 15 planes in general position in projective 3-space admits hyperbolic deformations.
Source: http://arxiv.org/abs/math/0202266v1
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Sep 21, 2013
09/13
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Bernard Shiffman; Steve Zelditch
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The Newton polytope $P_f$ of a polynomial $f$ is well known to have a strong impact on its zeros, as in the Kouchnirenko-Bernstein theorem on the number of simultaneous zeros of $m$ polynomials with given Newton polytopes. In this article, we show that $P_f$ also has a strong impact on the distribution of zeros of one or several polynomials. We equip the space of (holomorphic) polynomials of degree $\leq N$ in $m$ complex variables with its usual $SU(m+1)$-invariant Gaussian measure and then...
Source: http://arxiv.org/abs/math/0203074v2
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Sep 18, 2013
09/13
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Bernard Shiffman; Mikhail Zaidenberg
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We construct two classes of singular Kobayashi hyperbolic surfaces in $P^3$. The first consists of generic projections of the cartesian square $V = C \times C$ of a generic genus $g \ge 2$ curve $C$ smoothly embedded in $P^5$. These surfaces have C-hyperbolic normalizations; we give some lower bounds for their degrees and provide an example of degree 32. The second class of examples of hyperbolic surfaces in $P^3$ is provided by generic projections of the symmetric square $V' = C_2$ of a...
Source: http://arxiv.org/abs/math/9811152v1
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Sep 18, 2013
09/13
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Bernard Shiffman; Steve Zelditch
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We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers $L^N$ 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 $\{S^N_j\}$ of $H^0(M, L^N)$, we show that for almost every sequence $\{S^N_j\}$, the associated sequence of zero currents $1/N Z_{S^N_j}$ tends to the...
Source: http://arxiv.org/abs/math/9803052v1
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Sep 21, 2013
09/13
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Bernard Shiffman; Steve Zelditch
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We introduce several notions of `random fewnomials', i.e. random polynomials with a fixed number f of monomials of degree N. The f exponents are chosen at random and then the coefficients are chosen to be Gaussian random, mainly from the SU(m + 1) ensemble. The results give limiting formulas as N goes to infinity for the expected distribution of complex zeros of a system of k random fewnomials in m variables. When k = m, for SU(m + 1) polynomials, the limit is the Monge-Ampere measure of a...
Source: http://arxiv.org/abs/1011.3492v1
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Sep 18, 2013
09/13
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Bernard Shiffman; Mikhail Zaidenberg
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We show that a general small deformation of the union of two general cones in P3 of degree >= 4 is Kobayashi hyperbolic. Hence we obtain new examples of hyperbolic surfaces in P3 of any given degree d>= 8.
Source: http://arxiv.org/abs/math/0306360v1
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Sep 19, 2013
09/13
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Bernard Shiffman; Steve Zelditch
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We consider ensembles of random polynomials of the form $p(z)=\sum_{j = 1}^N a_j P_j$ where $\{a_j\}$ are independent complex normal random variables and where $\{P_j\}$ are the orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain $\Omega \subset C$ relative to an analytic weight $\rho(z) |dz|$. In the simplest case where $\Omega$ is the unit disk and $\rho=1$, so that $P_j(z) = z^j$, it is known that the average distribution of zeros is the uniform...
Source: http://arxiv.org/abs/math/0206162v1
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Sep 22, 2013
09/13
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Bernard Shiffman; Steve Zelditch; Scott Zrebiec
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We give asymptotic large deviations estimates for the volume inside a domain U of the zero set of a random polynomial of degree N, or more generally, of a holomorphic section of the N-th power of a positive line bundle on a compact Kaehler manifold. In particular, we show that for all $\delta>0$, the probability that this volume differs by more than $\delta N$ from its average value is less than $\exp(-C_{\delta,U}N^{m+1})$, for some constant $C_{\delta,U}>0$. As a consequence, the...
Source: http://arxiv.org/abs/0805.2598v2
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Sep 22, 2013
09/13
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Pavel Bleher; Bernard Shiffman; Steve Zelditch
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In our previous work [math-ph/9904020], we proved that the correlation functions for simultaneous zeros of random generalized polynomials have universal scaling limits and we gave explicit formulas for pair correlations in codimensions 1 and 2. The purpose of this paper is to compute these universal limits in all dimensions and codimensions. First, we use a supersymmetry method to express the n-point correlations as Berezin integrals. Then we use the Wick method to give a closed formula for the...
Source: http://arxiv.org/abs/math-ph/0011016v2
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Sep 23, 2013
09/13
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Pavel Bleher; Bernard Shiffman; Steve Zelditch
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We study the limit as $N\to\infty$ of the correlations between simultaneous zeros of random sections of the powers $L^N$ 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....
Source: http://arxiv.org/abs/math-ph/9904020v1
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Jul 19, 2013
07/13
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Bernard Shiffman; Steve Zelditch; Qi Zhong
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We study the conditional distribution of zeros of a Gaussian system of random polynomials (and more generally, holomorphic sections), given that the polynomials or sections vanish at a point p (or a fixed finite set of points). The conditional distribution is analogous to the pair correlation function of zeros, but we show that it has quite a different small distance behavior. In particular, the conditional distribution does not exhibit repulsion of zeros in dimension one. To prove this, we...
Source: http://arxiv.org/abs/1005.4166v1
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Sep 18, 2013
09/13
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Bernard Shiffman; Tatsuya Tate; Steve Zelditch
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We determine the asymptotics of the joint eigenfunctions of the torus action on a toric Kahler variety. Such varieties are models of completely integrable systems in complex geometry. We first determine the pointwise asymptotics of the eigenfunctions, which show that they behave like Gaussians centered at the corresponding classical torus. We then show that there is a universal Gaussian scaling limit of the distribution function near its center. We also determine the limit distribution for the...
Source: http://arxiv.org/abs/math/0306189v1
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Sep 18, 2013
09/13
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Pavel Bleher; Bernard Shiffman; Steve Zelditch
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This article is concerned with random holomorphic polynomials and their generalizations to algebraic and symplectic geometry. A natural algebro-geometric generalization studied in our prior work involves random holomorphic sections $H^0(M,L^N)$ of the powers of any positive line bundle $L \to M$ over any complex manifold. Our main interest is in the statistics of zeros of $k$ independent sections (generalized polynomials) of degree $N$ as $N\to\infty$. We fix a point $P$ and focus on the ball...
Source: http://arxiv.org/abs/math-ph/0002039v2
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Sep 18, 2013
09/13
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Pavel Bleher; Bernard Shiffman; Steve Zelditch
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This note is concerned with the scaling limit as N approaches infinity 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 L^N of any positive holomorphic line bundle L over a compact Kahler 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...
Source: http://arxiv.org/abs/math-ph/9903012v1
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Sep 20, 2013
09/13
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Jean-Pierre Demailly; Laszlo Lempert; Bernard Shiffman
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It is shown that every holomorphic map $f$ from a Runge domain $\Omega$ of an affine algebraic variety $S$ into a projective algebraic manifold $X$ is a uniform limit of Nash algebraic maps $f_\nu$ defined over an exhausting sequence of relatively compact open sets $\Omega_\nu$ in $\Omega$. A relative version is also given: If there is an algebraic subvariety $A$ (not necessarily reduced) in $S$ such that the restriction of $f$ to $A\cap\Omega$ is algebraic, then $f_\nu$ can be taken to...
Source: http://arxiv.org/abs/alg-geom/9212001v3
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Sep 21, 2013
09/13
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Michael R. Douglas; Bernard Shiffman; Steve Zelditch
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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. The present paper initiates the study of the statistics of critical points $\nabla s = 0$ of Gaussian random holomorphic sections with respect to a connection $\nabla$. Even the...
Source: http://arxiv.org/abs/math/0402326v2
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Sep 20, 2013
09/13
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Michael R. Douglas; Bernard Shiffman; Steve Zelditch
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A 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 and Denef-Douglas are given, together with van der Corput style remainder estimates. We also...
Source: http://arxiv.org/abs/math-ph/0506015v4
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Jul 20, 2013
07/13
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Michael R. Douglas; Bernard Shiffman; Steve Zelditch
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Motivated by the vacuum selection problem of string/M theory, we study a new geometric invariant of a positive Hermitian line bundle $(L, h)\to M$ over a compact K\"ahler manifold: the expected distribution of critical points of a Gaussian random holomorphic section $s \in H^0(M, L)$ with respect to the Chern connection $\nabla_h$. It is a measure on $M$ whose total mass is the average number $\mathcal{N}^{crit}_h$ of critical points of a random holomorphic section. We are interested in...
Source: http://arxiv.org/abs/math/0406089v3
