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

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21.
  • 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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22.
  • 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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23.
  • 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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24.
  • 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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25.
  • Berman, Robert, 1976 (författare)
  • On the strict convexity of the K-energy
  • 2019
  • Ingår i: Pure and Applied Mathematics Quarterly. - 1558-8599 .- 1558-8602. ; 15:4, s. 983-999
  • Tidskriftsartikel (refereegranskat)abstract
    • Let (X, L) be a polarized projective complex manifold. We show, by a simple toric one-dimensional example, that Mabuchi's K-energy functional on the geodesically complete space of bounded positive (1, 1)-forms in c(1)(L), endowed with the Mabuchi-Donaldson-Semmes metric, is not strictly convex modulo automorphisms. However, under some further assumptions the strict convexity in question does hold in the toric case. This leads to a uniqueness result saying that a finite energy minimizer of the K-energy (which exists on any toric polarized manifold (X, L) which is uniformly K-stable) is uniquely determined modulo automorphisms under the assumption that there exists some minimizer with strictly positive curvature current.
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26.
  • Berman, Robert, 1976, et al. (författare)
  • Propagation of chaos, wasserstein gradient flows and toric Kähler-Einstein metrics
  • 2018
  • Ingår i: Analysis and PDE. - : Mathematical Sciences Publishers. - 2157-5045 .- 1948-206X. ; 11:6, s. 1343-1380
  • Tidskriftsartikel (refereegranskat)abstract
    • Motivated by a probabilistic approach to Kähler-Einstein metrics we consider a general nonequilibrium statistical mechanics model in Euclidean space consisting of the stochastic gradient flow of a given (possibly singular) quasiconvex N-particle interaction energy. We show that a deterministic "macroscopic" evolution equation emerges in the large N-limit of many particles. This is a strengthening of previous results which required a uniform two-sided bound on the Hessian of the interaction energy. The proof uses the theory of weak gradient flows on the Wasserstein space. Applied to the setting of permanental point processes at "negative temperature", the corresponding limiting evolution equation yields a driftdiffusion equation, coupled to the Monge-Ampère operator, whose static solutions correspond to toric Kähler-Einstein metrics. This drift-diffusion equation is the gradient flow on the Wasserstein space of probability measures of the K-energy functional in Kähler geometry and it can be seen as a fully nonlinear version of various extensively studied dissipative evolution equations and conservation laws, including the Keller-Segel equation and Burger's equation. In a companion paper, applications to singular pair interactions in one dimension are given.
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27.
  • Berman, Robert, 1976, et al. (författare)
  • Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties
  • 2013
  • Ingår i: Annales de la faculté des sciences de Toulouse. - : Cellule MathDoc/CEDRAM. - 0240-2963 .- 2258-7519. ; 22:4, s. 649-711
  • Tidskriftsartikel (refereegranskat)abstract
    • We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in $\mathbb{R}^{n}$ with exponential non-linearity and target a convex body $P$ is solvable iff $0$ is the barycenter of $P.$ Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties $(X,\Delta )$ saying that $(X,\Delta )$ admits a (singular) Kähler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and extend to the log Fano setting the seminal result of Wang-Zhou concerning the case when $X$ is smooth and $\Delta $ is trivial. Li’s toric formula for the greatest lower bound on the Ricci curvature is also generalized. More generally, we obtain Kähler-Ricci solitons on any log Fano variety and show that they appear as the large time limit of the Kähler-Ricci flow. Furthermore, using duality, we also confirm a conjecture of Donaldson concerning solutions to Abreu’s boundary value problem on the convex body $P$ in the case of a given canonical measure on the boundary of $P.$
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28.
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29.
  • Berman, Robert, 1976 (författare)
  • Relative Kahler-Ricci flows and their quantization
  • 2010
  • Ingår i: preprint på arxiv.org.
  • 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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30.
  • Berman, Robert, 1976 (författare)
  • RELATIVE KAHLER-RICCI FLOWS AND THEIR QUANTIZATION
  • 2013
  • Ingår i: Analysis & Pde. - : Mathematical Sciences Publishers. - 1948-206X .- 2157-5045. ; 6:1, s. 131-180
  • Tidskriftsartikel (refereegranskat)abstract
    • Let pi : x -> S be a holomorphic fibration 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 and study their convergence properties. Mainly three different settings are investigated: the case when the fibers are Calabi-Yau manifolds and the case when L = +/- K-X/S is the relative ( anti) canonical line bundle. The main theme studied is whether "positivity in families" is preserved under the flows and its relation to the variation of the moduli of the complex structures of the fibers. 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 pi L-* -> S. Applications to the construction of canonical metrics on the relative canonical bundles of canonically polarized families 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 Kahler-Ricci flow of Donaldson's iteration in a certain double scaling limit.
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  • Resultat 21-30 av 41

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