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Sökning: LAR1:gu > Tidskriftsartikel > Chalmers tekniska högskola > Berman Robert 1976

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1.
  • Berman, Robert, 1976 (författare)
  • A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics
  • 2013
  • Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 248, s. 1254-1297
  • Tidskriftsartikel (refereegranskat)abstract
    • We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kahler manifold X. This functional can be seen as a generalization of Mabuchi's K-energy functional and its twisted versions to more singular situations. Applications to Monge-Ampere equations of mean field type, twisted Kahler-Einstein metrics and Moser-Trudinger type inequalities on Miller manifolds are given. Tian's alpha-invariant is generalized to singular measures, allowing in particular a proof of the existence of Kahler-Einstein metrics with positive Ricci curvature that are singular along a given anti-canonical divisor (which combined with very recent developments concerning Miller metrics with conical singularities confirms a recent conjecture of Donaldson). As another application we show that if the Calabi flow in the (anti-)canonical class exists for all times then it converges to a Kahler-Einstein metric, when a unique one exists, which is in line with a well-known conjecture. (C) 2013 Elsevier Inc. All rights reserved.
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2.
  • Berman, Robert, 1976, et al. (författare)
  • A variational approach to complex Monge-Ampere equations
  • 2013
  • Ingår i: Publications mathématiques. - : Springer Science and Business Media LLC. - 0073-8301. ; 117:1, s. 179-245
  • Tidskriftsartikel (refereegranskat)abstract
    • We show that degenerate complex Monge-Ampère equations in a big cohomology class of a compact Kähler manifold can be solved using a variational method, without relying on Yau’s theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kähler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kähler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kähler-Einstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as k→∞ of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy.
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3.
  • Berman, Robert, 1976, et al. (författare)
  • An arithmetic Hilbert-Samuel theorem for singular hermitian line bundles and cusp forms
  • 2014
  • Ingår i: Compositio Mathematica. - : Wiley. - 0010-437X .- 1570-5846. ; 150:10, s. 1703-1728
  • Tidskriftsartikel (refereegranskat)abstract
    • We prove arithmetic Hilbert-Samuel type theorems for semi-positive singular hermitian line bundles of finite height. This includes the log-singular metrics of Burgos-Kramer-Kuhn. The results apply in particular to line bundles of modular forms on some non-compact Shimura varieties. As an example, we treat the case of Hilbert modular surfaces, establishing an arithmetic analogue of the classical result expressing the dimensions of spaces of cusp forms in terms of special values of Dedekind zeta functions.
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4.
  • Berman, Robert, 1976 (författare)
  • Analytic torsion, vortices and positive Ricci curvature
  • 2010
  • Ingår i: preprint på arxiv.org.
  • Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
    • We characterize the global maximizers of a certain non-local functional defined on the space of all positively curved metrics on an ample line bundle L over a Kahler manifold X. This functional is an adjoint version, introduced by Berndtsson, of Donaldson's L-functional and generalizes the Ding-Tian functional whose critical points are Kahler-Einstein metrics of positive Ricci curvature. Applications to (1) analytic torsions on Fano manifolds (2) Chern-Simons-Higgs vortices on tori and (3) Kahler geometry are given. In particular, proofs of conjectures of (1) Gillet-Soul\'e and Fang (concerning the regularized determinant of Dolbeault Laplacians on the two-sphere) (2) Tarantello and (3) Aubin (concerning Moser-Trudinger type inequalities) in these three settings are obtained. New proofs of some results in Kahler geometry are also obtained, including a lower bound on Mabuchi's K-energy and the uniqueness result for Kahler-Einstein metrics on Fano manifolds of Bando-Mabuchi. This paper is a substantially extended version of the preprint arXiv:0905.4263 which it supersedes.
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6.
  • Berman, Robert, 1976, et al. (författare)
  • Bergman Geodesics
  • 2012
  • Ingår i: Lecture notes in mathematics. - Berlin, Heidelberg : Springer Berlin Heidelberg. - 0075-8434 .- 1617-9692. ; 2038, s. 283-302
  • Tidskriftsartikel (refereegranskat)abstract
    • The aim of this survey is to review the results of Phong-Sturm and Berndtsson on the convergence of Bergman geodesics towards geodesic segments in the space of positively curved metrics on an ample line bundle. As previously shown by Mabuchi, Semmes and Donaldson the latter geodesics may be described as solutions to the Dirichlet problem for a homogeneous complex Monge-Ampere equation. We emphasize in particular the relation between the convergence of the Bergman geodesics and semi-classical asymptotics for Berezin-Toeplitz quantization. Some extension to Wess-Zumino-Witten type equations are also briefly discussed.
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8.
  • Berman, Robert, 1976 (författare)
  • Bergman kernels and equilibrium measures for line bundles over projective manifolds
  • 2009
  • Ingår i: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 131:5, s. 1485-1524
  • 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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9.
  • Berman, Robert, 1976 (författare)
  • Bergman kernels and equilibrium measures for polarized pseudoconcave domains
  • 2010
  • Ingår i: International Journal of Mathematics. - 0129-167X. ; 21:1, s. 77-115
  • Tidskriftsartikel (refereegranskat)abstract
    • Let X be a domain in a closed polarized complex manifold (Y, L), where L is a (semi-) positive line bundle over Y. Any given Hermitian metric on L induces by restriction to X a Hilbert space structure on the space of global holomorphic sections on Y with values in the k-th tensor power of L (also using a volume form omega(n) on X). In this paper the leading large k asymptotics for the corresponding Bergman kernels and metrics are obtained in the case when X is a pseudo-concave domain with smooth boundary (under a certain compatibility assumption). The asymptotics are expressed in terms of the curvature of L and the boundary of X. The convergence of the Bergman metrics is obtained in a more general setting where (X, omega(n)) is replaced by any measure satisfying a Bernstein-Markov property. As an application the (generalized) equilibrium measure of the polarized pseudo-concave domain X is computed explicitly. Applications to the zero and mass distribution of random holomorphic sections and the eigenvalue distribution of Toeplitz operators will be described elsewhere.
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10.
  • Berman, Robert, 1976 (författare)
  • Bergman kernels for weighted polynomials and weighted equilibrium measures of C^n
  • 2009
  • Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 58:4, s. 1921-1946
  • 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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