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- Berman, Robert, 1976
(author)
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Analytic torsion, vortices and positive Ricci curvature
- 2010
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In: preprint på arxiv.org.
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Journal article (other academic/artistic)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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- Berman, Robert, 1976, et al.
(author)
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Convergence of Bergman measures of high powers of a line bundle
- 2008
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Other publication (other academic/artistic)abstract
- Let L be a holomorphic line bundle on a compact complex manifold X of dimension n, and let exp(-\phi) be a continuous metric on L. Fixing a measure dμ on X gives a sequence of Hilbert spaces consisting of holomorphic sections of tensor powers of L. We prove that the corresponding sequence of scaled Bergman measures converges, in the high tensor power limit, to the equilibrium measure of the pair (K,\phi), where K is the support of dμ, as long as dμ is stably Bernstein-Markov with respect to (K,\phi). Here the Bergman measure denotes dμ times the restriction to the diagonal of the pointwise norm of the corresponding orthogonal projection operator. In particular, an extension to higher dimensions is obtained of results concerning random matrices and classical orthogonal polynomials.
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- Berman, Robert, 1976, et al.
(author)
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Equidistribution of Fekete points on complex manifolds
- 2008
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In: www.arxiv.org, artikelnr 0807.0035.
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Journal article (other academic/artistic)abstract
- We prove the several variable version of the classical equidistribution theorem for Fekete points of a compact subset of the complex plane, which settles a well-known conjecture in pluri-potential theory. The result is obtained as a special case of a general equidistribution theorem for Fekete points in the setting of a given holomorphic line bundle over a compact complex manifold. The proof builds on our recent work "Capacities and weighted volumes for line bundles".
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- Berman, Robert, 1976, et al.
(author)
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Fekete points and convergence towards equilibrium measures on complex manifolds
- 2011
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In: Acta Mathematica. - : International Press of Boston. - 1871-2509 .- 0001-5962. ; 207:1, s. 1-27
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Journal article (other academic/artistic)abstract
- Building on the first two authors' previous results, we prove a general criterion for convergence of (possibly singular) Bergman measures towards equilibrium measures on complex manifolds. The criterion may be formulated in terms of growth properties of balls of holomorphic sections, or equivalently as an asymptotic minimization of generalized Donaldson L-functionals. Our result yields in particular the proof of a well-known conjecture in pluripotential theory concerning the equidistribution of Fekete points, and it also gives the convergence of Bergman measures towards equilibrium for Bernstein-Markov measures. Applications to interpolation of holomorphic sections are also discussed.
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