1. 
 Berman, Robert, 1976
(författare)

A thermodynamical formalism for MongeAmpere equations, MoserTrudinger inequalities and KahlerEinstein metrics
 2013

Ingår i: Advances in Mathematics.  00018708. ; 248, s. 12541297

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 Kenergy functional and its twisted versions to more singular situations. Applications to MongeAmpere equations of mean field type, twisted KahlerEinstein metrics and MoserTrudinger type inequalities on Miller manifolds are given. Tian's alphainvariant is generalized to singular measures, allowing in particular a proof of the existence of KahlerEinstein metrics with positive Ricci curvature that are singular along a given anticanonical 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 KahlerEinstein metric, when a unique one exists, which is in line with a wellknown conjecture. (C) 2013 Elsevier Inc. All rights reserved.


2. 
 Berman, Robert, 1976, et al.
(författare)

A variational approach to complex MongeAmpere equations
 2013

Ingår i: Publications mathématiques.  00738301. ; 117:1, s. 179245

Tidskriftsartikel (refereegranskat)abstract
 We show that degenerate complex MongeAmpère 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 KählerEinstein 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 DingTian’s variational characterization and BandoMabuchi’s uniqueness result to singular KählerEinstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as k→∞ of kbalanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy.


3. 


4. 
 Berman, Robert, 1976, et al.
(författare)

Bergman Geodesics
 2012

Ingår i: Lecture notes in mathematics.  00758434.  97836422366939783642236686 ; 2038, s. 283302

Tidskriftsartikel (refereegranskat)abstract
 The aim of this survey is to review the results of PhongSturm 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 MongeAmpere equation. We emphasize in particular the relation between the convergence of the Bergman geodesics and semiclassical asymptotics for BerezinToeplitz quantization. Some extension to WessZuminoWitten type equations are also briefly discussed.


5. 
 Berman, Robert, 1976
(författare)

Bergman kernels and equilibrium measures for line bundles over projective manifolds
 2009

Ingår i: American Journal of Mathematics, Volume 131, Number 5, October 2009. ; s. 14851524

Tidskriftsartikel (refereegranskat)abstract
 Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric h on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k th tensor power of L. In this paper various convergence results are obtained for the corresponding Bergman kernels (i.e. orthogonal projection kernels). The convergence is studied in the large k limit and is expressed in terms of the equilibrium metric h_e associated to h, as well as in terms of the MongeAmpere measure of h on a certain support set. It is also shown that the equilibrium metric h_e is in the class C^{1,1} on the complement of the augmented base locus of L. For L ample these results give generalizations of wellknown results concerning the case when the curvature of h is globally positive (then h_e=h). In general, the results can be seen as local metrized versions of Fujita's approximation theorem for the volume of L.


6. 
 Berman, Robert, 1976
(författare)

Bergman kernels and equilibrium measures for polarized pseudoconcave domains
 2010

Ingår i: International Journal of Mathematics.  0129167X. ; 21:1, s. 77

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 kth 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 pseudoconcave 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 BernsteinMarkov property. As an application the (generalized) equilibrium measure of the polarized pseudoconcave 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.


7. 
 Berman, Robert, 1976
(författare)

Bergman kernels for weighted polynomials and weighted equilibrium measures of C^n
 2009

Ingår i: Indiana Univ. Math. J. 58 (2009). ; s. 19211946

Tidskriftsartikel (refereegranskat)abstract
 Various convergence results for the Bergman kernel of the Hilbert space of all polynomials in \C^{n} of total degree at most k, equipped with a weighted norm, are obtained. The weight function is assumed to be C^{1,1}, i.e. it is differentiable and all of its first partial derivatives are locally Lipshitz continuous. The convergence is studied in the large k limit and is expressed in terms of the global equilibrium potential associated to the weight function, as well as in terms of the MongeAmpere measure of the weight function itself on a certain set. A setting of polynomials associated to a given Newton polytope, scaled by k, is also considered. These results apply directly to the study of the distribution of zeroes of random polynomials and of the eigenvalues of random normal matrices.


8. 
 Berman, Robert, 1976
(författare)

Determinantal Point Processes and Fermions on Complex Manifolds: Large Deviations and Bosonization AN G, 2000, LECT MATH ETH ZURICH
 2014

Ingår i: Communications in Mathematical Physics.  00103616. ; 327:1, s. 147

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 MongeAmpSre 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, 179245, 2013). We also express the LDP in terms of the RaySinger 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 bosonfermion correspondence.


9. 
 Berman, Robert, 1976, et al.
(författare)

Growth of balls of holomorphic sections and energy at equilibrium
 2010

Ingår i: Inventiones Mathematicae.  00209910. ; 181:2, s. 337394

Tidskriftsartikel (refereegranskat)abstract
 Let L be a big line bundle on a compact complex manifold X. Given a nonpluripolar 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 MongeAmpSre 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 supnorm 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 Robintype 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.


10. 
 Berman, Robert, 1976
(författare)

Holomorphic Morse inequalities on manifolds with boundary
 2005

Ingår i: Annales De L Institut Fourier.  03730956. ; 55:4, s. 1055

Tidskriftsartikel (refereegranskat)abstract
 Let X be a compact complex manifold with boundary and let Lk 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.

