| 11. |
- Berman, Robert, 1976
(författare)
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Convergence Rates for Discretized Monge-Ampere Equations and Quantitative Stability of Optimal Transport
- 2021
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Ingår i: Foundations of Computational Mathematics. - : Springer Science and Business Media LLC. - 1615-3375 .- 1615-3383. ; 21, s. 1099-1140
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Tidskriftsartikel (refereegranskat)abstract
- In recent works-both experimental and theoretical-it has been shown how to use computational geometry to efficiently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by discretizing one of the measures. Here we provide a quantitative convergence analysis for the solutions of the corresponding discretized Monge-Ampere equations. This yields H-1-converge rates, in terms of the corresponding spatial resolution h, of the discrete approximations of the optimal transport map, when the source measure is discretized and the target measure has bounded convex support. Periodic variants of the results are also established. The proofs are based on new quantitative stability results for optimal transport maps, shown using complex geometry.
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| 12. |
- Berman, Robert, 1976, et al.
(författare)
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Convexity of the extended K-energy and the large time behavior of the weak Calabi flow
- 2017
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Ingår i: Geometry and Topology. - : Mathematical Sciences Publishers. - 1465-3060 .- 1364-0380. ; 21:5, s. 2945-2988
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Tidskriftsartikel (refereegranskat)abstract
- © 2017, Mathematical Sciences Publishers. All rights reserved. Let (X, ω) be a compact connected Kähler manifold and denote by(ε p , d p ) the metric completion of the space of Kähler potentials H ω with respect to the L p -type path length metric d p . First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to ε p is a d p -1sc functional that is convex along finite-energy geodesics. Second, following the program of J Streets, we use this to study the asymptotics of the weak (twisted) Calabi flow inside the CAT(0) metric space (ε 2 , d 2 ). This flow exists for all times and coincides with the usual smooth (twisted) Calabi flow whenever the latter exists. We show that the weak (twisted) Calabi flow either diverges with respect to the d 2 -metric or it d 1 -converges to some minimizer of the K-energy inside ε 2 . This gives the first concrete result about the long-time convergence of this flow on general Kähler manifolds, partially confirming a conjecture of Donaldson. We investigate the possibility of constructing destabilizing geodesic rays asymptotic to diverging weak (twisted) Calabi trajectories, and give a result in the case when the twisting form is Kähler. Finally, when a cscK metric exists in H ω , our results imply that the weak Calabi flow d 1 -converges to such a metric.
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| 13. |
- Berman, Robert, 1976, et al.
(författare)
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Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics
- 2017
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Ingår i: Journal of the American Mathematical Society. - : American Mathematical Society (AMS). - 0894-0347 .- 1088-6834. ; 30:4, s. 1165-1196
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Tidskriftsartikel (refereegranskat)abstract
- We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kähler potentials on a compact Kähler manifold, thus confirming a conjecture of Chen, and give some applications in Kähler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogeneous Monge-Ampère equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.
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| 14. |
- Berman, Robert, 1976
(författare)
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Determinantal Point Processes and Fermions on Complex Manifolds: Large Deviations and Bosonization
- 2014
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 327:1, s. 1-47
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Tidskriftsartikel (refereegranskat)abstract
- We study determinantal random point processes on a compact complex manifold X associated to a Hermitian metric on a line bundle over X and a probability measure on X. Physically, this setup describes a gas of free fermions on X subject to a U(1)-gauge field and when X is the Riemann sphere it specializes to various random matrix ensembles. Our general setup will also include the setting of weighted orthogonal polynomials in , as well as in . It is shown that, in the many particle limit, the empirical random measures on X converge exponentially towards the deterministic pluripotential equilibrium measure, defined in terms of the Monge-AmpSre operator of complex pluripotential theory. More precisely, a large deviation principle (LDP) is established with a good rate functional which coincides with the (normalized) pluricomplex energy of a measure recently introduced in Berman et al. (Publ Math de l'IHA parts per thousand S 117, 179-245, 2013). We also express the LDP in terms of the Ray-Singer analytic torsion. This can be seen as an effective bosonization formula, generalizing the previously known formula in the Riemann surface case to higher dimensions and the paper is concluded with a heuristic quantum field theory interpretation of the resulting effective boson-fermion correspondence.
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| 15. |
- 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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| 16. |
- 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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| 17. |
- Berman, Robert, 1976
(författare)
-
From Monge–Ampère equations to envelopes and geodesic rays in the zero temperature limit
- 2019
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Ingår i: Mathematische Zeitschrift. - : Springer Science and Business Media LLC. - 1432-1823 .- 0025-5874. ; 291:1-2, s. 365-394
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Tidskriftsartikel (refereegranskat)abstract
- Let (X, θ) be a compact complex manifold X equipped with a smooth (but not necessarily positive) closed (1, 1)-form θ. By a well-known envelope construction this data determines, in the case when the cohomology class [θ] is pseudoeffective, a canonical θ-psh function u θ . When the class [θ] is Kähler we introduce a family u β of regularizations of u θ , parametrized by a large positive number β, where u β is defined as the unique smooth solution of a complex Monge–Ampère equation of Aubin–Yau type. It is shown that, as β→ ∞, the functions u β converge to the envelope u θ uniformly on X in the Hölder space C 1,α (X) for any α∈] 0 , 1 [(which is optimal in terms of Hölder exponents). A generalization of this result to the case of a nef and big cohomology class is also obtained and a weaker version of the result is obtained for big cohomology classes. The proofs of the convergence results do not assume any a priori regularity of u θ . Applications to the regularization of ω-psh functions and geodesic rays in the closure of the space of Kähler metrics are given. As briefly explained there is a statistical mechanical motivation for this regularization procedure, where β appears as the inverse temperature. This point of view also leads to an interpretation of u β as a “transcendental” Bergman metric.
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| 18. |
- Berman, Robert, 1976, et al.
(författare)
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Growth of balls of holomorphic sections and energy at equilibrium
- 2010
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 181:2, s. 337-394
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Tidskriftsartikel (refereegranskat)abstract
- Let L be a big line bundle on a compact complex manifold X. Given a non-pluripolar compact subset K of X and a continuous Hermitian metric e (-phi) on L, we define the energy at equilibrium of (K,phi) as the Monge-AmpSre energy of the extremal psh weight associated to (K,phi). We prove the differentiability of the energy at equilibrium with respect to phi, and we show that this energy describes the asymptotic behaviour as k -> a of the volume of the sup-norm unit ball induced by (K,k phi) on the space of global holomorphic sections H (0)(X,kL). As a consequence of these results, we recover and extend Rumely's Robin-type formula for the transfinite diameter. We also obtain an asymptotic description of the analytic torsion, and extend Yuan's equidistribution theorem for algebraic points of small height to the case of a big line bundle.
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| 19. |
- 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
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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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| 20. |
- Berman, Robert, 1976
(författare)
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K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
- 2016
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 203:3, s. 973-1025
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Tidskriftsartikel (refereegranskat)abstract
- It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof is based on a new formula expressing the Donaldson-Futaki invariants in terms of the slope of the Ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti-canonical line bundle of X. One consequence is that a toric Fano variety X is K-polystable iff it is K-polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope P associated to X. The results also extend to the logarithmic setting and in particular to the setting of Kahler-Einsteinmetrics with edge-cone singularities. Applications to geodesic stability, bounds on the Ricci potential and Perelman's lambda-entropy functional on K-unstable Fano manifolds are also given.
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