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- Berman, Robert, 1976
(författare)
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Bergman kernels and equilibrium measures for line bundles over projective manifolds
- 2009
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Ingår i: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 131:5, s. 1485-1524
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Tidskriftsartikel (refereegranskat)abstract
- Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric h on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k th tensor power of L. In this paper various convergence results are obtained for the corresponding Bergman kernels (i.e. orthogonal projection kernels). The convergence is studied in the large k limit and is expressed in terms of the equilibrium metric h_e associated to h, as well as in terms of the Monge-Ampere measure of h on a certain support set. It is also shown that the equilibrium metric h_e is in the class C^{1,1} on the complement of the augmented base locus of L. For L ample these results give generalizations of well-known results concerning the case when the curvature of h is globally positive (then h_e=h). In general, the results can be seen as local metrized versions of Fujita's approximation theorem for the volume of L.
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- Berman, Robert, 1976
(författare)
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Bergman kernels for weighted polynomials and weighted equilibrium measures of C^n
- 2009
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 58:4, s. 1921-1946
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Tidskriftsartikel (refereegranskat)abstract
- Various convergence results for the Bergman kernel of the Hilbert space of all polynomials in \C^{n} of total degree at most k, equipped with a weighted norm, are obtained. The weight function is assumed to be C^{1,1}, i.e. it is differentiable and all of its first partial derivatives are locally Lipshitz continuous. The convergence is studied in the large k limit and is expressed in terms of the global equilibrium potential associated to the weight function, as well as in terms of the Monge-Ampere measure of the weight function itself on a certain set. A setting of polynomials associated to a given Newton polytope, scaled by k, is also considered. These results apply directly to the study of the distribution of zeroes of random polynomials and of the eigenvalues of random normal matrices.
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- Berman, Robert, 1976
(författare)
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Holomorphic Morse inequalities on manifolds with boundary
- 2005
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Ingår i: Annales De L Institut Fourier. - 0373-0956. ; 55:4, s. 1055-
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Tidskriftsartikel (refereegranskat)abstract
- Let X be a compact complex manifold with boundary and let L-k be a high power of a hermitian holomorphic line bundle over X. When X has no boundary, Demailly's holomorphic Morse inequalities give asymptotic bounds on the dimensions of the Dolbeault cohomology groups with values in Lk, in terms of the curvature of L. We extend Demailly's inequalities to the case when X has a boundary by adding a boundary term expressed as a certain average of the curvature of the line bundle and the Levi curvature of the boundary. Examples are given that show that the inequalities are sharp.
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- Berman, Robert, 1976
(författare)
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Super toeplitz operators on line bundles
- 2006
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Ingår i: Journal of Geometric Analysis. - 1050-6926. ; 16:1, s. 1-22
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Tidskriftsartikel (refereegranskat)abstract
- Let Lk be a high power of a hermitian holomorphic line bundle over a complex manifold X. Given a differential form f on X, we define a super Toeplitz operator Tf acting on the space of harmonic (0, q)-forms with values in Lk, with symbol f. The asymptotic distribution of its eigenvalues, when k tends to infinity, is obtained in terms of the symbol of the operator and the curvature of the line bundle L, given certain conditions on the curvature. For example, already when q = 0, i.e., the case of holomorphic sections, this generalizes a result of Bautet de Monvel and Guillemin to semi-positive line bundles. The asympotics are obtained from the asymptotics of the Bergman kernels of the corresponding harmonic spaces, which have independent interest. Applications to sampling are also given.
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