| 1. |
- Berman, Robert, 1976-, et al.
(författare)
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A variational approach to complex Monge-Ampere equations
- 2012
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Ingår i: Publications mathématiques. - 0073-8301.
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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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| 2. |
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| 3. |
- Berman, Robert, 1976-
(författare)
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Bergman kernels and equilibrium measures for line bundles over projective manifolds
- 2009
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Ingår i: American Journal of Mathematics, Volume 131, Number 5, October 2009. ; s. 1485-1524
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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 Monge-Ampere 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 well-known 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.
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| 4. |
- Berman, Robert, 1976-
(författare)
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BERGMAN KERNELS AND EQUILIBRIUM MEASURES FOR POLARIZED PSEUDO-CONCAVE DOMAINS
- 2010
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Ingår i: International Journal of Mathematics. - 0129-167X. ; 21:1, s. 77
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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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| 5. |
- Berman, Robert, 1976-
(författare)
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Bergman kernels for weighted polynomials and weighted equilibrium measures of C^n
- 2009
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Ingår i: Indiana Univ. Math. J. 58 (2009). ; s. 1921-1946
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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 Monge-Ampere 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.
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| 6. |
- 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. - 0020-9910. ; 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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| 7. |
- 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, s. 1055
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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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| 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. ; :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-
(författare)
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Sharp Asymptotics for Toeplitz Determinants and Convergence Towards the Gaussian Free Field on Riemann Surfaces
- 2012
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Ingår i: International Mathematics Research Notices. - 1073-7928. ; :22, s. 5031-5062
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Tidskriftsartikel (refereegranskat)abstract
- We consider canonical determinantal random point processes with N particles on a compact Riemann surface X defined with respect to the constant curvature metric. We establish strong exponential concentration of measure type properties involving Dirichlet norms of linear statistics. This gives an optimal central limit theorem (CLT), saying that the fluctuations of the corresponding empirical measures converge, in the large N limit, towards the Laplacian of the Gaussian free field on X in the strongest possible sense. The CLT is also shown to be equivalent to a new sharp strong Szego-type theorem for Toeplitz determinants in this context. One of the ingredients in the proofs are new Bergman kernel asymptotics providing exponentially small error terms in a constant curvature setting.
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| 10. |
- Berman, Robert, 1976-
(författare)
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Super toeplitz operators on line bundles
- 2006
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Ingår i: Journal of Geometric Analysis. - 10506926. ; 16:1, s. 1-22
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Tidskriftsartikel (refereegranskat)abstract
- Let Lk be a high power of a hermitian holomorphic line bundle over a complex manifold X. Given a differential form f on X, we define a super Toeplitz operator Tf acting on the space of harmonic (0, q)-forms with values in Lk, with symbol f. The asymptotic distribution of its eigenvalues, when k tends to infinity, is obtained in terms of the symbol of the operator and the curvature of the line bundle L, given certain conditions on the curvature. For example, already when q = 0, i.e., the case of holomorphic sections, this generalizes a result of Bautet de Monvel and Guillemin to semi-positive line bundles. The asympotics are obtained from the asymptotics of the Bergman kernels of the corresponding harmonic spaces, which have independent interest. Applications to sampling are also given.
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