Catalogue et commandes en ligne (paiement sécurisé, VISA ou MASTERCARD uniquement)

Revues disponibles par abonnement

Annales scientifiques de l'ENS

Astérisque

Bulletin de la SMF

Mémoires de la SMF

Revue d'Histoire des Mathématiques

Gazette des Mathématiciens

Séries de livres

Astérisque

Cours Spécialisés

Documents Mathématiques

Mémoires de la SMF

Panoramas et Synthèses

Séminaires et Congrès

Série Chaire Jean Morlet

SMF/AMS Texts and Monographs

La Série T

Fascicules « Journée Annuelle »

Autres livres

Donald E. Knuth - traductions françaises

Rééditions du Séminaire Nicolas Bourbaki

Rééditions des Œuvres de Jean Leray

Revue de l'Institut Elie Cartan

Editions électroniques

Annales scientifiques de l'ENS

Bulletin de la SMF

Revue d'Histoire des Mathématiques

Séminaires et Congrès

Plus d'information / Abonnement

Publications grand public

L'explosion des mathématiques (smf.emath.fr)

Mathématiques L'explosion continue (smf.emath.fr)

Zoom sur les métiers des maths (smf.emath.fr)

Zoom sur les métiers des mathématiques et de l'informatique (smf.emath.fr)

Où en sont les mathématiques ?

La Série T

Pour les auteurs

Soumission des manuscrits

Formats et documentation

Plus d'info

Liste de diffusion électronique (smf.emath.fr)

Information pour les libraires et diffuseurs (smf.emath.fr)

Publications de la SMF
fr en
Votre numéro IP : 54.80.219.236
Accès aux édit. élec. : SémCong

Annales scientifiques de l'ENS

Présentation de la publication

Parutions

Dernières parutions

Comité de rédaction / Secrétariat

Série 4 :
Série 3 :
Série 2 :
Série 1 :

Faire une recherche


Catalogue & commande

Annales scientifiques de l'ENS - Parutions - série 4, 48 (2015)

Parutions < série 4, 48

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 48, fascicule 3 (2015)

Dan Coman, George Marinescu
Equidistribution results for singular metrics on line bundles
Annales scientifiques de l'ENS 48, fascicule 3 (2015), 497-536

Télécharger cet article : Fichier PDF
Acheter l'ouvrage

Résumé :
Résultats d'équidistribution pour métriques singulières sur des fibrés en droites
Considérons un fibré holomorphe en droites L muni d'une métrique singulière h au-dessus d'une variété complexe X. Soit _p le courant de Fubini-Study associé naturellement à l'espace des sections holomorphes de carré intégrable de L^p. En supposant que le lieu singulier de la métrique h est contenu dans un ensemble analytique compact X tel que k et que le logarithme du noyau de Bergman associé à L^p_X a l'ordre de croissance o(p), p, nous prouvons que : 1) Les courants _p^k convergent faiblement sur X vers c_1(L,h)^k, où c_1(L,h) est le courant de courbure de h. 2) Les moyennes des zéros communs d'un k-vecteur aléatoire de sections holomphes L^2-intégrables convergent faiblement dans le sens des courants vers c_1(L,h)^k. L'hypothèse de croissance du noyau de Bergman est la conséquence de son développement asymptotique dans le cas d'une métrique lisse h. Nous la démontrons ici sous des conditions assez générales. Nous montrons ensuite que nos résultats s'appliquent à nombre de situations géométriques (métriques singulières sur un fibré gros, métriques de Kähler-Einstein sur des ouverts de Zariski, quotients arithmétiques...).

Mots-clefs : Noyau de Bergman, courants de Fubini-Study, métrique hermitienne singulière, équidistribution des zéros, sections holomorphes aléatoires.

Abstract:
Let (L,h) be a holomorphic line bundle with a positively curved singular Hermitian metric over a complex manifold X. One can define naturally the sequence of Fubini-Study currents _p associated to the space of L^2-holomorphic sections of L^p. Assuming that the singular set of the metric is contained in a compact analytic subset  of X and that the logarithm of the Bergman density function of L^p_X grows like o(p) as p, we prove the following: 1) the currents _p^k converge weakly on the whole X to c_1(L,h)^k, where c_1(L,h) is the curvature current of h. 2) the expectations of the common zeros of a random k-tuple of L^2-holomorphic sections converge weakly in the sense of currents to c_1(L,h)^k. Here k is so that k. Our weak asymptotic condition on the Bergman density function is known to hold in many cases, as it is a consequence of its asymptotic expansion. We also prove it here in a quite general setting. We then show that many important geometric situations (singular metrics on big line bundles, Kähler-Einstein metrics on Zariski-open sets, arithmetic quotients) fit into our framework.

Keywords: Bergman density function, Fubini-Study currents, singular Hermitian metric, equidistribution of zeros, random holomorphic sections.

Class. math. : 32L10; 32U40, 32W20, 53C55


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Andreotti, Aldo and Grauert, Hans
Théorème de finitude pour la cohomologie des espaces complexes
Bull. Soc. Math. France 90 (1962) 193–259
Math Reviews MR0150342 (27 \#343)
2
Ash, Avner and Mumford, David and Rapoport, Michael and Tai, Yung-Sheng
Smooth compactifications of locally symmetric varieties
Cambridge Univ. Press, Cambridge, 2010
Math Reviews MR2590897 (2010m:14067)
3
Bedford, Eric and Taylor, B. A.
The Dirichlet problem for a complex Monge-Ampère equation
Invent. Math. 37 (1976) 1–44
Math Reviews MR0445006 (56 \#3351)
4
Bedford, Eric and Taylor, B. A.
A new capacity for plurisubharmonic functions
Acta Math. 149 (1982) 1–40
Math Reviews MR674165 (84d:32024)
5
Berman, Robert J.
Bergman kernels and equilibrium measures for line bundles over projective manifolds
Amer. J. Math. 131 (2009) 1485–1524
Math Reviews MR2559862 (2010g:32030)
6
Berman, Robert J. and Berndtsson, Bo and Sjöstrand, Johannes
A direct approach to Bergman kernel asymptotics for positive line bundles
Ark. Mat. 46 (2008) 197–217
Math Reviews MR2430724 (2009k:58050)
7
Berman, Robert J. and Boucksom, Sébastien
Growth of balls of holomorphic sections and energy at equilibrium
Invent. Math. 181 (2010) 337–394
Math Reviews MR2657428 (2011h:32021)
8
Bierstone, Edward and Milman, Pierre D.
Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant
Invent. Math. 128 (1997) 207–302
Math Reviews MR1440306 (98e:14010)
9
Błocki, Zbigniew
On the definition of the Monge-Ampère operator in C2
Math. Ann. 328 (2004) 415–423
Math Reviews MR2036329 (2004m:32074)
10
Błocki, Zbigniew
The domain of definition of the complex Monge-Ampère operator
Amer. J. Math. 128 (2006) 519–530
Math Reviews MR2214901 (2006k:32076)
11
Catlin, David
The Bergman kernel and a theorem of Tian
in Analysis and geometry in several complex variables (Katata, 1997)
Trends Math. (1999) 1–23
Math Reviews MR1699887 (2000e:32001)
12
Cegrell, Urban
The general definition of the complex Monge-Ampère operator
Ann. Inst. Fourier (Grenoble) 54 (2004) 159–179
Math Reviews MR2069125 (2005d:32062)
13
Colţoiu, Mihnea
On the Oka-Grauert principle for 1-convex manifolds
Math. Ann. 310 (1998) 561–569
Math Reviews MR1612254 (99c:32040)
14
Coman, Dan and Guedj, Vincent and Zeriahi, Ahmed
Domains of definition of Monge-Ampère operators on compact Kähler manifolds
Math. Z. 259 (2008) 393–418
Math Reviews MR2390088 (2009a:32053)
15
Coman, Dan and Marinescu, George
Convergence of Fubini-study currents for orbifold line bundles
Internat. J. Math. 24 (2013) 1350051
Math Reviews MR3084732
16
Coman, Dan and Marinescu, George
On the approximation of positive closed currents on compact Kähler manifolds
Math. Rep. (Bucur.) 15 (2013) 373–386
Math Reviews MR3237479
17
Dai, Xianzhe and Liu, Kefeng and Ma, Xiaonan
On the asymptotic expansion of Bergman kernel
J. Differential Geom. 72 (2006) 1–41
Math Reviews MR2215454 (2007k:58043)
18
Demailly, Jean-Pierre
Estimations L2 pour l'opérateur d'un fibré vectoriel holomorphe semi-positif au-dessus d'une variété kählérienne complète
Ann. Sci. École Norm. Sup. 15 (1982) 457–511
Math Reviews MR690650 (85d:32057)
19
Demailly, Jean-Pierre
Singular Hermitian metrics on positive line bundles
in Complex algebraic varieties (Bayreuth, 1990)
Lecture Notes in Math. 1507 (1992) 87–104
Math Reviews MR1178721 (93g:32044)
20
Demailly, Jean-Pierre
Regularization of closed positive currents and intersection theory
J. Algebraic Geom. 1 (1992) 361–409
Math Reviews MR1158622 (93e:32015)
21
Demailly, Jean-Pierre
Monge-Ampère operators, Lelong numbers and intersection theory
in Complex analysis and geometry
Univ. Ser. Math. (1993) 115–193
Math Reviews MR1211880 (94k:32009)
22
Demailly, Jean-Pierre
A numerical criterion for very ample line bundles
J. Differential Geom. 37 (1993) 323–374
Math Reviews MR1205448 (94d:14007)
23
Dinh, Tien-Cuong and Marinescu, George and Schmidt, Viktoria
Equidistribution of zeros of holomorphic sections in the non-compact setting
J. Stat. Phys. 148 (2012) 113–136
Math Reviews MR2950760
24
Dinh, Tien-Cuong and Sibony, Nessim
Geometry of currents, intersection theory and dynamics of horizontal-like maps
Ann. Inst. Fourier (Grenoble) 56 (2006) 423–457
Math Reviews MR2226022 (2008c:32047)
25
Dinh, Tien-Cuong and Sibony, Nessim
Distribution des valeurs de transformations méromorphes et applications
Comment. Math. Helv. 81 (2006) 221–258
Math Reviews MR2208805 (2007i:32017)
26
Federer, Herbert
Geometric measure theory
Springer, 1969
Math Reviews MR0257325 (41 \#1976)
27
Folland, Gerald B.
Real analysis
John Wiley Sons, Inc., New York, 1999
Math Reviews MR1681462 (2000c:00001)
28
Folland, Gerald B. and Kohn, J. J.
The Neumann problem for the Cauchy-Riemann complex
Princeton Univ. Press, Princeton, N.J., 1972
Math Reviews MR0461588 (57 \#1573)
29
Fornæss, John Erik and Sibony, Nessim
Oka's inequality for currents and applications
Math. Ann. 301 (1995) 399–419
Math Reviews MR1324517 (96k:32013)
30
Fornæss, John Erik and Sibony, Nessim
Complex dynamics in higher dimension. II
in Modern methods in complex analysis (Princeton, NJ, 1992)
Ann. of Math. Stud. 137 (1995) 135–182
Math Reviews MR1369137 (97g:32033)
31
Gauduchon, Paul
Le théorème de l'excentricité nulle
C. R. Acad. Sci. Paris Sér. A-B 285 (1977) A387–A390
Math Reviews MR0470920 (57 \#10664)
32
Guedj, Vincent and Zeriahi, Ahmed
Intrinsic capacities on compact Kähler manifolds
J. Geom. Anal. 15 (2005) 607–639
Math Reviews MR2203165 (2006j:32041)
33
Guedj, Vincent and Zeriahi, Ahmed
The weighted Monge-Ampère energy of quasiplurisubharmonic functions
J. Funct. Anal. 250 (2007) 442–482
Math Reviews MR2352488 (2008h:32056)
34
Gunning, Robert C. and Rossi, Hugo
Analytic functions of several complex variables
AMS Chelsea Publishing, Providence, RI, 2009
Math Reviews MR2568219 (2010j:32001)
35
Harvey, Reese
Holomorphic chains and their boundaries
in Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975)
(1977) 309–382
Math Reviews MR0447619 (56 \#5929)
36
Holowinsky, Roman and Soundararajan, Kannan
Mass equidistribution for Hecke eigenforms
Ann. of Math. 172 (2010) 1517–1528
Math Reviews MR2680499 (2011i:11061)
37
Hörmander, Lars
Notions of convexity
Birkhäuser, 2007
Math Reviews MR2311920
38
Hsiao, Chin-Yu and Marinescu, George
Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles
Comm. Anal. Geom. 22 (2014) 1–108
Math Reviews MR3194375
39
Ji, Shanyu and Shiffman, Bernard
Properties of compact complex manifolds carrying closed positive currents
J. Geom. Anal. 3 (1993) 37–61
Math Reviews MR1197016 (93m:32014)
40
Kobayashi, Ryoichi
Kähler-Einstein metric on an open algebraic manifold
Osaka J. Math. 21 (1984) 399–418
Math Reviews MR752470 (85i:53053)
41
Lelong, Pierre and Gruman, Lawrence
Entire functions of several complex variables
Springer, 1986
Math Reviews MR837659 (87j:32001)
42
Ma, Xiaonan and Marinescu, George
Generalized Bergman kernels on symplectic manifolds
C. R. Math. Acad. Sci. Paris 339 (2004) 493–498
Math Reviews MR2099548 (2005k:58065)
43
Ma, Xiaonan and Marinescu, George
Holomorphic Morse inequalities and Bergman kernels
Birkhäuser, 2007
Math Reviews MR2339952 (2008g:32030)
44
Ma, Xiaonan and Marinescu, George
Generalized Bergman kernels on symplectic manifolds
Adv. Math. 217 (2008) 1756–1815
Math Reviews MR2382740 (2009c:58047)
45
Marinescu, George and Dinh, Tien-Cuong
On the compactification of hyperconcave ends and the theorems of Siu-Yau and Nadel
Invent. Math. 164 (2006) 233–248
Math Reviews MR2218480 (2007b:32033b)
46
Marshall, Simon
Mass equidistribution for automorphic forms of cohomological type on GL2
J. Amer. Math. Soc. 24 (2011) 1051–1103
Math Reviews MR2813338 (2012i:11049)
47
Mumford, David
Hirzebruch's proportionality theorem in the noncompact case
Invent. Math. 42 (1977) 239–272
Math Reviews MR471627 (81a:32026)
48
Nadel, Alan
On complex manifolds which can be compactified by adding finitely many points
Invent. Math. 101 (1990) 173–189
Math Reviews MR1055714 (91g:32036)
49
Ohsawa, Takeo and Takegoshi, Kenshō
On the extension of L2 holomorphic functions
Math. Z. 195 (1987) 197–204
Math Reviews MR892051 (88g:32029)
50
Ruan, Wei-Dong
Canonical coordinates and Bergmann metrics
Comm. Anal. Geom. 6 (1998) 589–631
Math Reviews MR1638878 (2000a:32050)
51
Rudnick, Zeév
On the asymptotic distribution of zeros of modular forms
Int. Math. Res. Not. 2005 (2005) 2059–2074
Math Reviews MR2181743 (2006k:11099)
52
Rudnick, Zeév and Sarnak, Peter
The behaviour of eigenstates of arithmetic hyperbolic manifolds
Comm. Math. Phys. 161 (1994) 195–213
Math Reviews MR1266075 (95m:11052)
53
Shiffman, Bernard
Convergence of random zeros on complex manifolds
Sci. China Ser. A 51 (2008) 707–720
Math Reviews MR2395415 (2009b:32007)
54
Shiffman, Bernard and Zelditch, Steve
Distribution of zeros of random and quantum chaotic sections of positive line bundles
Comm. Math. Phys. 200 (1999) 661–683
Math Reviews MR1675133 (2001j:32018)
55
Shiffman, Bernard and Zelditch, Steve
Random polynomials with prescribed Newton polytope
J. Amer. Math. Soc. 17 (2004) 49–108
Math Reviews MR2015330 (2005e:60032)
56
Shiffman, Bernard and Zelditch, Steve
Number variance of random zeros on complex manifolds
Geom. Funct. Anal. 18 (2008) 1422–1475
Math Reviews MR2465693 (2009k:32019)
57
Sibony, Nessim
Quelques problèmes de prolongement de courants en analyse complexe
Duke Math. J. 52 (1985) 157–197
Math Reviews MR791297 (87e:32020)
58
Siu, Yum Tong
Analyticity of sets associated to Lelong numbers and the extension of closed positive currents
Invent. Math. 27 (1974) 53–156
Math Reviews MR0352516 (50 \#5003)
59
Tian, Gang
On a set of polarized Kähler metrics on algebraic manifolds
J. Differential Geom. 32 (1990) 99–130
Math Reviews MR1064867 (91j:32031)
60
To, Wing-Keung
Distribution of zeros of sections of canonical line bundles over towers of covers
J. London Math. Soc. 63 (2001) 387–399
Math Reviews MR1810136 (2002c:32029)
61
Yau, Shing Tung
A general Schwarz lemma for Kähler manifolds
Amer. J. Math. 100 (1978) 197–203
Math Reviews MR0486659 (58 \#6370)
62
Zelditch, Steve
Szegő kernels and a theorem of Tian
Int. Math. Res. Not. 1998 (1998) 317–331
Math Reviews MR1616718 (99g:32055)