| 1. |
- Berman, Robert, 1976
(författare)
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A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics
- 2013
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Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 248, s. 1254-1297
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Tidskriftsartikel (refereegranskat)abstract
- We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kahler manifold X. This functional can be seen as a generalization of Mabuchi's K-energy functional and its twisted versions to more singular situations. Applications to Monge-Ampere equations of mean field type, twisted Kahler-Einstein metrics and Moser-Trudinger type inequalities on Miller manifolds are given. Tian's alpha-invariant is generalized to singular measures, allowing in particular a proof of the existence of Kahler-Einstein metrics with positive Ricci curvature that are singular along a given anti-canonical divisor (which combined with very recent developments concerning Miller metrics with conical singularities confirms a recent conjecture of Donaldson). As another application we show that if the Calabi flow in the (anti-)canonical class exists for all times then it converges to a Kahler-Einstein metric, when a unique one exists, which is in line with a well-known conjecture. (C) 2013 Elsevier Inc. All rights reserved.
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| 2. |
- Berman, Robert, 1976, et al.
(författare)
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A variational approach to complex Monge-Ampere equations
- 2013
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Ingår i: Publications mathématiques. - : Springer Science and Business Media LLC. - 0073-8301. ; 117:1, s. 179-245
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Tidskriftsartikel (refereegranskat)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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| 3. |
- Berman, Robert, 1976, et al.
(författare)
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An arithmetic Hilbert-Samuel theorem for singular hermitian line bundles and cusp forms
- 2014
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Ingår i: Compositio Mathematica. - : Wiley. - 0010-437X .- 1570-5846. ; 150:10, s. 1703-1728
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Tidskriftsartikel (refereegranskat)abstract
- We prove arithmetic Hilbert-Samuel type theorems for semi-positive singular hermitian line bundles of finite height. This includes the log-singular metrics of Burgos-Kramer-Kuhn. The results apply in particular to line bundles of modular forms on some non-compact Shimura varieties. As an example, we treat the case of Hilbert modular surfaces, establishing an arithmetic analogue of the classical result expressing the dimensions of spaces of cusp forms in terms of special values of Dedekind zeta functions.
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| 4. |
- Berman, Robert, 1976, et al.
(författare)
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Bergman Geodesics
- 2012
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Ingår i: Lecture notes in mathematics. - Berlin, Heidelberg : Springer Berlin Heidelberg. - 0075-8434 .- 1617-9692. ; 2038, s. 283-302
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Tidskriftsartikel (refereegranskat)abstract
- The aim of this survey is to review the results of Phong-Sturm and Berndtsson on the convergence of Bergman geodesics towards geodesic segments in the space of positively curved metrics on an ample line bundle. As previously shown by Mabuchi, Semmes and Donaldson the latter geodesics may be described as solutions to the Dirichlet problem for a homogeneous complex Monge-Ampere equation. We emphasize in particular the relation between the convergence of the Bergman geodesics and semi-classical asymptotics for Berezin-Toeplitz quantization. Some extension to Wess-Zumino-Witten type equations are also briefly discussed.
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| 5. |
- Berman, Robert, 1976
(författare)
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Bergman kernels and equilibrium measures for polarized pseudoconcave domains
- 2010
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Ingår i: International Journal of Mathematics. - 0129-167X. ; 21:1, s. 77-115
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Tidskriftsartikel (refereegranskat)abstract
- Let X be a domain in a closed polarized complex manifold (Y, L), where L is a (semi-) positive line bundle over Y. Any given Hermitian metric on L induces by restriction to X a Hilbert space structure on the space of global holomorphic sections on Y with values in the k-th tensor power of L (also using a volume form omega(n) on X). In this paper the leading large k asymptotics for the corresponding Bergman kernels and metrics are obtained in the case when X is a pseudo-concave domain with smooth boundary (under a certain compatibility assumption). The asymptotics are expressed in terms of the curvature of L and the boundary of X. The convergence of the Bergman metrics is obtained in a more general setting where (X, omega(n)) is replaced by any measure satisfying a Bernstein-Markov property. As an application the (generalized) equilibrium measure of the polarized pseudo-concave domain X is computed explicitly. Applications to the zero and mass distribution of random holomorphic sections and the eigenvalue distribution of Toeplitz operators will be described elsewhere.
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| 6. |
- 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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| 7. |
- 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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| 8. |
- Berman, Robert, 1976
(författare)
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Kahler-Einstein metrics emerging from free fermions and statistical mechanics
- 2011
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Ingår i: Journal of High Energy Physics. - 1126-6708 .- 1029-8479. ; :10
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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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| 9. |
- Berman, Robert, 1976, et al.
(författare)
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Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs
- 2014
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Ingår i: Geometric and Functional Analysis. - : Springer Science and Business Media LLC. - 1016-443X .- 1420-8970. ; 24:6, s. 1683-1730
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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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| 10. |
- Berman, Robert, 1976, et al.
(författare)
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Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties
- 2013
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Ingår i: Annales de la faculté des sciences de Toulouse. - : Cellule MathDoc/CEDRAM. - 0240-2963 .- 2258-7519. ; 22:4, s. 649-711
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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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