Equidistribution results for singular metrics on line bundles

Coman, Dan; Marinescu, George

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 3, pp. 497-536.

Considérons un fibré holomorphe en droites $L$ muni d'une métrique singulière $h$ au-dessus d'une variété complexe $X$. Soit ${\gamma }_p$ le courant de Fubini-Study associé naturellement à l'espace des sections holomorphes de carré intégrable de $L^{\otimes p}$. En supposant que le lieu singulier de la métrique $h$ est contenu dans un ensemble analytique compact $\Sigma \subset X$ tel que $codim\Sigma \ge k$ et que le logarithme du noyau de Bergman associé à $L^{\otimes p}{\upharpoonright }_{X\setminus \Sigma }$ a l'ordre de croissance $o\left(p\right)$, $p\to \infty$, nous prouvons que :

1) Les courants ${\gamma }_p^k$ 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...).

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 ${\gamma }_p$ associated to the space of $L^2$-holomorphic sections of $L^{\otimes p}$. Assuming that the singular set of the metric is contained in a compact analytic subset $\Sigma$ of $X$ and that the logarithm of the Bergman density function of $L^{\otimes p}{\upharpoonright }_{X\setminus \Sigma }$ grows like $o\left(p\right)$ as $p\to \infty$, we prove the following:

1) the currents ${\gamma }_p^k$ 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 $codim\Sigma \ge k$. 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.

Classification : 32L10; 32U40, 32W20, 53C55
Keywords: Bergman density function, Fubini-Study currents, singular Hermitian metric, equidistribution of zeros, random holomorphic sections.
Mot clés : Noyau de Bergman, courants de Fubini-Study, métrique hermitienne singulière, équidistribution des zéros, sections holomorphes aléatoires.

Andreotti, A.; Grauert, H. Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, Volume 90 (1962), pp. 193-259 (ISSN: 0037-9484) | DOI | Numdam | MR | Zbl

Ash, A.; Mumford, D.; Rapoport, M.; Tai, Y.-S., Cambridge Mathematical Library, Cambridge Univ. Press, Cambridge, 2010, 230 pages (ISBN: 978-0-521-73955-9) | DOI | MR | Zbl

Berman, R. J.; Boucksom, S. Growth of balls of holomorphic sections and energy at equilibrium, Invent. Math., Volume 181 (2010), pp. 337-394 (ISSN: 0020-9910) | DOI | MR | Zbl

Berman, R. J.; Berndtsson, B.; Sjöstrand, J. A direct approach to Bergman kernel asymptotics for positive line bundles, Ark. Mat., Volume 46 (2008), pp. 197-217 (ISSN: 0004-2080) | DOI | MR | Zbl

Berman, R. J. Bergman kernels and equilibrium measures for line bundles over projective manifolds, Amer. J. Math., Volume 131 (2009), pp. 1485-1524 (ISSN: 0002-9327) | DOI | MR | Zbl

Bierstone, E.; Milman, P. D. Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math., Volume 128 (1997), pp. 207-302 (ISSN: 0020-9910) | DOI | MR | Zbl

Bedford, E.; Taylor, B. A. The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., Volume 37 (1976), pp. 1-44 (ISSN: 0020-9910) | DOI | MR | Zbl

Bedford, E.; Taylor, B. A. A new capacity for plurisubharmonic functions, Acta Math., Volume 149 (1982), pp. 1-40 (ISSN: 0001-5962) | DOI | MR | Zbl

Błocki, Z. On the definition of the Monge-Ampère operator in ${ℂ}^2$ , Math. Ann., Volume 328 (2004), pp. 415-423 (ISSN: 0025-5831) | DOI | MR | Zbl

Błocki, Z. The domain of definition of the complex Monge-Ampère operator, Amer. J. Math., Volume 128 (2006), pp. 519-530 http://muse.jhu.edu/journals/american_journal_of_mathematics/v128/128.2bl_ocki.pdf (ISSN: 0002-9327) | DOI | MR | Zbl

Catlin, D., Analysis and geometry in several complex variables (Katata, 1997) (Trends Math.), Birkhäuser, 1999, pp. 1-23 | MR | Zbl

Cegrell, U. The general definition of the complex Monge-Ampère operator, Ann. Inst. Fourier (Grenoble), Volume 54 (2004), pp. 159-179 http://aif.cedram.org/item?id=AIF_2004__54_1_159_0 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Coman, D.; Guedj, V.; Zeriahi, A. Domains of definition of Monge-Ampère operators on compact Kähler manifolds, Math. Z., Volume 259 (2008), pp. 393-418 (ISSN: 0025-5874) | DOI | MR | Zbl

Coman, D.; Marinescu, G. Convergence of Fubini-study currents for orbifold line bundles, Internat. J. Math., Volume 24 (2013), 1350051 pages (ISSN: 0129-167X) | DOI | MR | Zbl

Coman, D.; Marinescu, G. On the approximation of positive closed currents on compact Kähler manifolds, Math. Rep. (Bucur.), Volume 15 (2013), pp. 373-386 (ISSN: 1582-3067) | MR | Zbl

Colţoiu, M. On the Oka-Grauert principle for 1-convex manifolds, Math. Ann., Volume 310 (1998), pp. 561-569 (ISSN: 0025-5831) | DOI | MR | Zbl

Demailly, J.-P. Estimations $L^2$ pour l'opérateur $\overline{\partial }$ d'un fibré vectoriel holomorphe semi-positif au-dessus d'une variété kählérienne complète, Ann. Sci. École Norm. Sup., Volume 15 (1982), pp. 457-511 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Demailly, J.-P. Regularization of closed positive currents and intersection theory, J. Algebraic Geom., Volume 1 (1992), pp. 361-409 (ISSN: 1056-3911) | MR | Zbl

Demailly, J.-P., Complex algebraic varieties (Bayreuth, 1990) (Lecture Notes in Math.), Volume 1507, Springer, 1992, pp. 87-104 | DOI | MR | Zbl

Demailly, J.-P. A numerical criterion for very ample line bundles, J. Differential Geom., Volume 37 (1993), pp. 323-374 http://projecteuclid.org/euclid.jdg/1214453680 (ISSN: 0022-040X) | MR | Zbl

Demailly, J.-P., Complex analysis and geometry (Univ. Ser. Math.), Plenum, New York, 1993, pp. 115-193 | DOI | MR | Zbl

Dai, X.; Liu, K.; Ma, X. On the asymptotic expansion of Bergman kernel, J. Differential Geom., Volume 72 (2006), pp. 1-41 http://projecteuclid.org/euclid.jdg/1143593124 (ISSN: 0022-040X) | MR | Zbl

Dinh, T.-C.; Marinescu, G.; Schmidt, V. Equidistribution of zeros of holomorphic sections in the non-compact setting, J. Stat. Phys., Volume 148 (2012), pp. 113-136 (ISSN: 0022-4715) | DOI | MR | Zbl

Dinh, T.-C.; Sibony, N. Distribution des valeurs de transformations méromorphes et applications, Comment. Math. Helv., Volume 81 (2006), pp. 221-258 (ISSN: 0010-2571) | DOI | MR | Zbl

Dinh, T.-C.; Sibony, N. Geometry of currents, intersection theory and dynamics of horizontal-like maps, Ann. Inst. Fourier (Grenoble), Volume 56 (2006), pp. 423-457 http://aif.cedram.org/item?id=AIF_2006__56_2_423_0 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Federer, H., Grundl. math. Wiss., 153, Springer, 1969, 676 pages | MR | Zbl

Folland, G. B.; Kohn, J. J., Ann. of Math. Studies, 75, Princeton Univ. Press, Princeton, N.J., 1972, 146 pages | MR | Zbl

Folland, G. B., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999, 386 pages (ISBN: 0-471-31716-0) | MR | Zbl

Fornæss, J. E.; Sibony, N., Modern methods in complex analysis (Princeton, NJ, 1992) (Ann. of Math. Stud.), Volume 137, Princeton Univ. Press, Princeton, NJ, 1995, pp. 135-182 | MR | Zbl

Fornæss, J. E.; Sibony, N. Oka's inequality for currents and applications, Math. Ann., Volume 301 (1995), pp. 399-419 (ISSN: 0025-5831) | DOI | MR | Zbl

Gauduchon, P. Le théorème de l'excentricité nulle, C. R. Acad. Sci. Paris Sér. A-B, Volume 285 (1977), p. A387-A390 | MR | Zbl

Gunning, R. C.; Rossi, H., AMS Chelsea Publishing, Providence, RI, 2009, 318 pages (ISBN: 978-0-8218-2165-7) | MR | Zbl

Guedj, V.; Zeriahi, A. Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal., Volume 15 (2005), pp. 607-639 (ISSN: 1050-6926) | DOI | MR | Zbl

Guedj, V.; Zeriahi, A. The weighted Monge-Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal., Volume 250 (2007), pp. 442-482 (ISSN: 0022-1236) | DOI | MR | Zbl

Harvey, R., Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975), Amer. Math. Soc., Providence, R. I., 1977, pp. 309-382 | MR | Zbl

Hsiao, C.-Y.; Marinescu, G. Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles, Comm. Anal. Geom., Volume 22 (2014), pp. 1-108 (ISSN: 1019-8385) | DOI | MR | Zbl

Hörmander, L., Modern Birkhäuser Classics, Birkhäuser, 2007, 414 pages (ISBN: 978-0-8176-4584-7; 0-8176-4584-5) | MR | Zbl

Holowinsky, R.; Soundararajan, K. Mass equidistribution for Hecke eigenforms, Ann. of Math., Volume 172 (2010), pp. 1517-1528 (ISSN: 0003-486X) | DOI | MR | Zbl

Ji, S.; Shiffman, B. Properties of compact complex manifolds carrying closed positive currents, J. Geom. Anal., Volume 3 (1993), pp. 37-61 (ISSN: 1050-6926) | DOI | MR | Zbl

Kobayashi, R. Kähler-Einstein metric on an open algebraic manifold, Osaka J. Math., Volume 21 (1984), pp. 399-418 http://projecteuclid.org/euclid.ojm/1200777118 (ISSN: 0030-6126) | MR | Zbl

Lelong, P.; Gruman, L., Grundl. math. Wiss., 282, Springer, 1986, 270 pages (ISBN: 3-540-15296-2) | DOI | MR | Zbl

Marshall, S. Mass equidistribution for automorphic forms of cohomological type on $G{L}_2$ , J. Amer. Math. Soc., Volume 24 (2011), pp. 1051-1103 (ISSN: 0894-0347) | DOI | MR | Zbl

Marinescu, G.; Dinh, T.-C. On the compactification of hyperconcave ends and the theorems of Siu-Yau and Nadel, Invent. Math., Volume 164 (2006), pp. 233-248 (ISSN: 0020-9910) | DOI | MR | Zbl

Ma, X.; Marinescu, G. Generalized Bergman kernels on symplectic manifolds, C. R. Math. Acad. Sci. Paris, Volume 339 (2004), pp. 493-498 (ISSN: 1631-073X) | DOI | MR | Zbl

Ma, X.; Marinescu, G., Progress in Math., 254, Birkhäuser, 2007, 422 pages (ISBN: 978-3-7643-8096-0) | MR | Zbl

Ma, X.; Marinescu, G. Generalized Bergman kernels on symplectic manifolds, Adv. Math., Volume 217 (2008), pp. 1756-1815 (ISSN: 0001-8708) | DOI | MR | Zbl

Mumford, D. Hirzebruch's proportionality theorem in the noncompact case, Invent. Math., Volume 42 (1977), pp. 239-272 (ISSN: 0020-9910) | DOI | MR | Zbl

Nadel, A. On complex manifolds which can be compactified by adding finitely many points, Invent. Math., Volume 101 (1990), pp. 173-189 (ISSN: 0020-9910) | DOI | MR | Zbl

Ohsawa, T.; Takegoshi, K. On the extension of $L^2$ holomorphic functions, Math. Z., Volume 195 (1987), pp. 197-204 (ISSN: 0025-5874) | DOI | MR | Zbl

Rudnick, Z.; Sarnak, P. The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys., Volume 161 (1994), pp. 195-213 http://projecteuclid.org/euclid.cmp/1104269797 (ISSN: 0010-3616) | DOI | MR | Zbl

Ruan, W.-D. Canonical coordinates and Bergmann metrics, Comm. Anal. Geom., Volume 6 (1998), pp. 589-631 (ISSN: 1019-8385) | DOI | MR | Zbl

Rudnick, Z. On the asymptotic distribution of zeros of modular forms, Int. Math. Res. Not., Volume 2005 (2005), pp. 2059-2074 (ISSN: 1073-7928) | DOI | MR | Zbl

Shiffman, B. Convergence of random zeros on complex manifolds, Sci. China Ser. A, Volume 51 (2008), pp. 707-720 (ISSN: 1006-9283) | DOI | MR | Zbl

Sibony, N. Quelques problèmes de prolongement de courants en analyse complexe, Duke Math. J., Volume 52 (1985), pp. 157-197 (ISSN: 0012-7094) | DOI | MR | Zbl

Siu, Y. T. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., Volume 27 (1974), pp. 53-156 (ISSN: 0020-9910) | DOI | MR | Zbl

Shiffman, B.; Zelditch, S. Random polynomials with prescribed Newton polytope, J. Amer. Math. Soc., Volume 17 (2004), pp. 49-108 (ISSN: 0894-0347) | DOI | MR | Zbl

Shiffman, B.; Zelditch, S. Number variance of random zeros on complex manifolds, Geom. Funct. Anal., Volume 18 (2008), pp. 1422-1475 (ISSN: 1016-443X) | DOI | MR | Zbl

Shiffman, B.; Zelditch, S. Distribution of zeros of random and quantum chaotic sections of positive line bundles, Comm. Math. Phys., Volume 200 (1999), pp. 661-683 (ISSN: 0010-3616) | DOI | MR | Zbl

Tian, G. On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom., Volume 32 (1990), pp. 99-130 http://projecteuclid.org/euclid.jdg/1214445039 (ISSN: 0022-040X) | MR | Zbl

To, W.-K. Distribution of zeros of sections of canonical line bundles over towers of covers, J. London Math. Soc., Volume 63 (2001), pp. 387-399 (ISSN: 0024-6107) | DOI | MR | Zbl

Yau, S. T. A general Schwarz lemma for Kähler manifolds, Amer. J. Math., Volume 100 (1978), pp. 197-203 (ISSN: 0002-9327) | DOI | MR | Zbl

Zelditch, S. Szegő kernels and a theorem of Tian, Int. Math. Res. Not., Volume 1998 (1998), pp. 317-331 (ISSN: 1073-7928) | DOI | MR | Zbl