| 21. |
- Berman, Robert, 1976
(författare)
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Analytic torsion, vortices and positive Ricci curvature
- 2010
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Ingår i: preprint på arxiv.org.
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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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| 22. |
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| 23. |
- Berman, Robert, 1976, et al.
(författare)
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Equidistribution of Fekete points on complex manifolds
- 2008
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Ingår i: www.arxiv.org, artikelnr 0807.0035.
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)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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| 24. |
- Berman, Robert, 1976, et al.
(författare)
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Fekete points and convergence towards equilibrium measures on complex manifolds
- 2011
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Ingår i: Acta Mathematica. - : International Press of Boston. - 1871-2509 .- 0001-5962. ; 207:1, s. 1-27
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)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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| 25. |
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| 26. |
- Berman, Robert, 1976
(författare)
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Relative Kahler-Ricci flows and their quantization
- 2010
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Ingår i: preprint på arxiv.org.
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- Let X be a complex manifold fibered over the base S and let L be a relatively ample line bundle over X. We define relative Kahler-Ricci flows on the space of all Hermitian metrics on L with relatively positive curvature. Mainly three different settings are investigated: the case when the fibers are Calabi-Yau manifolds and the case when L is the relative (anti-) canonical line bundle. The main theme studied is whether posivity in families is preserved under the flows and its relation to the variation of the moduli of the complex structures of the fibres. The quantization of this setting is also studied, where the role of the Kahler-Ricci flow is played by Donaldson's iteration on the space of all Hermitian metrics on the finite rank vector bundle over S defined as the zeroth direct image induced by the fibration. Applications to the construction of canonical metrics on relative canonical bumdles and Weil-Petersson geometry are given. Some of the main results are a parabolic analogue of a recent elliptic equation of Schumacher and the convergence towards the K\"ahler-Ricci flow of Donaldson's iteration in a certain double scaling limit.
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| 27. |
- Berman, Robert, 1976
(författare)
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Sharp asymptotics for Toeplitz determinants, fluctuations and the gaussian free field on a Riemann surface
- 2011
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Ingår i: preprint på arxiv.org.
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- We consider canonical determinantal random point processes with N particles on a compact Riemann surface X defined with respect to the constant curvature metric. In the higher genus (hyperbolic) cases these point processes may be defined in terms of automorphic forms. We establish strong exponential concentration of measure type properties involving Dirichlet norms of linear statistics. This gives an optimal Central Limit Theorem (CLT), saying that the fluctuations of the corresponding empirical measures converge, in the large N limt, towards the Laplacian of the Gaussian free field on X in the strongest possible sense. The CLT is also shown to be equivalent to a new sharp strong Szeg\"o type theorem for Toeplitz determinants in this context. One of the ingredients in the proofs are new Bergman kernel asymptotics providing exponentially small error terms in a constant curvature setting.
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