Finiteness of totally geodesic exceptional divisors in Hermitian locally symmetric spaces

Koziarz, Vincent; Maubon, Julien

Bulletin de la Société Mathématique de France, Volume 146 (2018) no. 4, pp. 613-631

We prove that on a smooth complex surface which is a compact quotient of the bidisc or of the 2-ball, there is at most a finite number of totally geodesic curves with negative self-intersection. More generally, there are only finitely many exceptional totally geodesic divisors in a compact Hermitian locally symmetric space of noncompact type of dimension at least 2. This is deduced from a convergence result for currents of integration along totally geodesic subvarieties in compact Hermitian locally symmetric spaces, which itself follows from an equidistribution theorem for totally geodesic submanifolds in a locally symmetric space of finite volume.

Nous prouvons que sur une surface complexe lisse qui est un quotient compact du bidisque ou de la boule de dimension 2, il n’y a qu’un nombre fini de courbes totalement géodésiques d’auto-intersection strictement négative. Plus généralement, il n’y a qu’un nombre fini de diviseurs totalement géodésiques exceptionnels dans une variété localement symétrique (de type non compact) hermitienne compacte de dimension au moins 2. Ces énoncés sont déduits d’un théorème de convergence de courants d’intégration le long de sous-variétés totalement géodésiques dans les variétés localement symétriques hermitiennes compactes, lui-même obtenu à partir d’un résultat d’équidistribution des sous-variétés totalement géodésiques dans les variétés localement symétriques de volume fini.

Classification: 53C35, 32M15, 22E40, 37C40, 37C85, 32C30, 14G35
Keywords: Bounded Negativity conjecture, Hermitian locally symmetric spaces, totally geodesic submanifold, equidistribution, negative curve, exceptional divisor, current of integration
Mots-clés : Conjecture de la négativité bornée, espaces localement symétriques hermitiens, sous-variété totalement géodésique, équidistribution, courbe d’auto-intersection négative, diviseur exceptionnel, courant d’intégration

[1] W. Barth, K. Hulek, C. Peters & A. Van de Ven, “Compact complex surfaces”, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 4, vol. 2, Springer, 2004. | Zbl

[2] T. Bauer, B. Harbourne, A. L. Knutsen, A. Küronya, S. Müller-Stach, X. Roulleau & T. Szemberg, “Negative curves on algebraic surfaces”, Duke Math. J. 162 (2013), p. 1877–1894. | MR | Zbl

[3] Y. Benoist & J.-F. Quint “Stationary measures and invariant subsets of homogeneous spaces (III)”, Ann. of Math. 178 (2013), p. 1017–1059. | MR | Zbl

[4] J. Berndt & C. Olmos, “On the index of symmetric spaces”, J. reine angew. Math., Ahead of Print DOI 10.1515/crelle-2015-0060, 2015. | MR

[5] A. Borel & Harish-Chandra, “Arithmetic subgroups of algebraic groups”, Ann. of Math. 75 (1962), p. 485–535. | MR | Zbl

[6] A. Borel & J. Tits, “Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I”, Invent. Math. 12 (1971), p. 95–104. | MR | Zbl

[7] M. Bridson & A. Haefliger, “Metric spaces of non-positive curvature”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer, 1999. | MR | Zbl

[8] L. Clozel & E. Ullmo, “Equidistribution de sous-variétés spéciales”, Ann. of Math. 161 (2005), p. 1571–1588. | MR | Zbl

[9] S. Di Rocco, A. Küronya, S. Müller-Stach & T. Szemberg, “Mini-Workshop: Negative Curves on Algebraic Surfaces”, Mathematisches Forschungsinstitut Oberwolfach, Report No. 10/2014, available at http://www.mfo.de/document/1409b/OWR_2014_10.pdf.

[10] A. Eskin & G. Margulis, “Recurrence properties of random walks on finite volume homogeneous manifolds”, Random walks and geometry, Walter de Gruyter, 2004, p. 431–444. | MR | Zbl

[11] A. Eskin, S. Mozes & N. Shah, “Unipotent flows and counting lattice points on homogeneous spaces”, Ann. of Math. 143 (1996), p. 253–299. | MR | Zbl

[12] S. Helgason, “Differential geometry, Lie groups, and symmetric spaces”, Corrected reprint of the 1978 original, Graduate Studies in Mathematics, 34, American Mathematical Society, 2001. | MR | Zbl

[13] A. Kollross, “Polar actions on symmetric spaces”, J. Differential Geom. 77 (2007), p. 425–482. | MR | Zbl

[14] G. A. Margulis, “Discrete subgroups of semisimple Lie groups”, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer, 1991. | MR | Zbl

[15] Y. Miyaoka, “The orbibundle Miyaoka-Yau-Sakai inequality and an effective Bogomolov-McQuillan theorem”, Publ. Res. Inst. Math. Sci. 44 (2008), p. 403–417. | MR | Zbl

[16] M. Möller & D. Toledo, “Bounded negativity of self-intersection numbers of Shimura curves in Shimura surfaces”, Algebra Number Theory 9 (2015), p. 897–912. | MR

[17] S. Mozes & N. Shah, “On the space of ergodic invariant measures of unipotent flows”, Ergodic Theory Dynam. Systems 15 (1995), p. 149–159. | MR | Zbl

[18] A. L. Onishchik, “Totally geodesic submanifolds of symmetric spaces” (Russian), in Geometric methods in problems of algebra and analysis no. 2 (Russian), Yaroslav. Gos. Univ., Yaroslavl’, 161 (1980), p. 64–85. | MR

[19] M. S. Raghunathan, “Discrete subgroups of Lie groups”, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 68. Springer, 1972. | MR | Zbl

[20] M. Ratner, “Invariant measures and orbit closures for unipotent actions on homogeneous spaces”, Geom. Funct. Anal. 4 (1994), p. 236–257. | MR | Zbl

[21] R. W. Richardson Jr., “A rigidity theorem for subalgebras of Lie and associative algebras”, Illinois J. Math. 11 (1967), p. 92–110. | MR | Zbl

[22] E. Ullmo, “Equidistribution de sous-variétés spéciales II”, J. Reine Angew. Math. 606 (2007), p. 193–216. | MR | Zbl

[23] V. S. Varadarajan, “Lie groups, Lie algebras, and their representations”, Reprint of the 1974 edition. Graduate Texts in Mathematics, vol. 102. Springer, 1984. | MR | Zbl

[24] A. Zeghib, “Ensembles invariants des flots géodésiques des variétés localement symétriques”, Ergodic Theory Dynam. Systems 15 (1995), p. 379–412. | MR | Zbl