| 1. |
- Berman, Robert, 1976, et al.
(author)
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A variational approach to complex Monge-Ampere equations
- 2013
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In: Publications mathématiques. - : Springer Science and Business Media LLC. - 0073-8301. ; 117:1, s. 179-245
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Journal article (peer-reviewed)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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| 2. |
- 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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| 3. |
- 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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