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

  • Resultat 11-20 av 34
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11.
  • Berman, Robert, 1976, et al. (författare)
  • Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics
  • 2017
  • Ingår i: Journal of the American Mathematical Society. - : American Mathematical Society (AMS). - 0894-0347 .- 1088-6834. ; 30:4, s. 1165-1196
  • 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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12.
  • Berman, Robert, 1976 (författare)
  • Determinantal Point Processes and Fermions on Complex Manifolds: Large Deviations and Bosonization
  • 2014
  • Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 327:1, s. 1-47
  • 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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13.
  • Berman, Robert, 1976 (författare)
  • From Monge–Ampère equations to envelopes and geodesic rays in the zero temperature limit
  • 2019
  • Ingår i: Mathematische Zeitschrift. - : Springer Science and Business Media LLC. - 1432-1823 .- 0025-5874. ; 291:1-2, s. 365-394
  • 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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14.
  • Berman, Robert, 1976, et al. (författare)
  • Growth of balls of holomorphic sections and energy at equilibrium
  • 2010
  • Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 181:2, s. 337-394
  • 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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15.
  • Berman, Robert, 1976 (författare)
  • Holomorphic Morse inequalities on manifolds with boundary
  • 2005
  • Ingår i: Annales De L Institut Fourier. - 0373-0956. ; 55:4
  • 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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16.
  • Berman, Robert, 1976 (författare)
  • K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
  • 2016
  • Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 203:3, s. 973-1025
  • 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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17.
  • Berman, Robert, 1976 (författare)
  • Kahler-Einstein metrics emerging from free fermions and statistical mechanics
  • 2011
  • Ingår i: Journal of High Energy Physics. - 1126-6708 .- 1029-8479. ; :10
  • Tidskriftsartikel (refereegranskat)abstract
    • We propose a statistical mechanical derivation of Kithler-Einstein metrics, i.e. solutions to Einstein's vacuum field equations in Euclidean signature (with a cosmological constant) on a compact Kahler manifold X. The microscopic theory is given by a canonical free fermion gas on X whose one-particle states are pluricanonical holomorphic sections on X (coinciding with higher spin states in the case of a Riemann surface) defined in background free manner. A heuristic, but hopefully physically illuminating, argument for the convergence in the thermodynamical (large N) limit is given, based on a recent mathematically rigorous result about exponentially small fluctuation's of Slater determinants. Relations to higher-dimensional effective bosonization, the Yau-Tian-Donaldson program in Kahler geometry and quantum gravity are explored. The precise mathematical details will be investigated elsewhere.
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18.
  • Berman, Robert, 1976, et al. (författare)
  • Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs
  • 2014
  • Ingår i: Geometric and Functional Analysis. - : Springer Science and Business Media LLC. - 1016-443X .- 1420-8970. ; 24:6, s. 1683-1730
  • Tidskriftsartikel (refereegranskat)abstract
    • Let X be a canonically polarized variety, i.e. a complex projective variety such that its canonical class KXdefines an ample (Formula presented.)-line bundle, and satisfying the conditions G1and S2. Our main result says that X admits a Kähler–Einstein metric iff X has semi-log canonical singularities i.e. iff X is a stable variety in the sense of Kollár–Shepherd-Barron and Alexeev (whose moduli spaces are known to be compact). By definition a Kähler–Einstein metric in this singular context simply means a Kähler–Einstein on the regular locus of X with volume equal to the algebraic volume of KX, i.e. the top intersection number of KX. We also show that such a metric is uniquely determined and extends to define a canonical positive current in c1(KX). Combined with recent results of Odaka our main result shows that X admits a Kähler–Einstein metric iff X is K-stable, which thus confirms the Yau–Tian–Donaldson conjecture in this general setting of (possibly singular) canonically polarized varieties. More generally, our results are shown to hold in the setting of log minimal varieties and they also generalize some prior results concerning Kähler–Einstein metrics on quasi-projective varieties.
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19.
  • Berman, Robert, 1976 (författare)
  • Large Deviations for Gibbs Measures with Singular Hamiltonians and Emergence of Kahler-Einstein Metrics
  • 2017
  • Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 354:3, s. 1133-1172
  • Tidskriftsartikel (refereegranskat)abstract
    • In the present paper and the companion paper (Berman, Kahler-Einstein metrics, canonical random point processes and birational geometry. arXiv:1307.3634, 2015) a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on complex algebraic varieties X is introduced by sampling "temperature deformed" determinantal point processes. The main new ingredient is a large deviation principle for Gibbs measures with singular Hamiltonians, which is proved in the present paper. As an application we show that the unique Kahler-Einstein metric with negative Ricci curvature on a canonically polarized algebraic manifold X emerges in the many particle limit of the canonical point processes on X. In the companion paper (Berman in 2015) the extension to algebraic varieties X with positive Kodaira dimension is given and a conjectural picture relating negative temperature states to the existence problem for Kahler-Einstein metrics with positive Ricci curvature is developed.
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20.
  • Berman, Robert, 1976 (författare)
  • On Large Deviations for Gibbs Measures, Mean Energy and Gamma-Convergence
  • 2018
  • Ingår i: Constructive Approximation. - : Springer Science and Business Media LLC. - 0176-4276 .- 1432-0940. ; 48:1, s. 3-30
  • Tidskriftsartikel (refereegranskat)abstract
    • We consider the random point processes on a measure space defined by the Gibbs measures associated with a given sequence of N-particle Hamiltonians Inspired by the method of Messer-Spohn for proving concentration properties for the laws of the corresponding empirical measures, we propose a number of hypotheses on that are quite general but still strong enough to extend the approach of Messer-Spohn. The hypotheses are formulated in terms of the asymptotics of the corresponding mean energy functionals. We show that in many situations, the approach even yields a large deviation principle (LDP) for the corresponding laws. Connections to Gamma-convergence of (free) energy type functionals at different levels are also explored. The focus is on differences between positive and negative temperature situations, motivated by applications to complex geometry. The results yield, in particular, large deviation principles at positive as well as negative temperatures for quite general classes of singular mean field models with pair interactions, generalizing the 2D vortex model and Coulomb gases. In a companion paper, the results will be illustrated in the setting of Coulomb and Riesz type gases on a Riemannian manifold X, comparing with the complex geometric setting.
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  • Resultat 11-20 av 34
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