Orbifold generic semi-positivity: an application to families of canonically polarized manifolds
[Semi-positivité orbifolde : une application aux familles de variétés canoniquement polarisées]
Annales de l'Institut Fourier, Tome 65 (2015) no. 2, pp. 835-861.

Nous définissons la notion de ‘fibré cotangent orbifolde’ Ω 1 (X,Δ) pour une paire (X,Δ) log-canonique : ce fibré est défini sur des revêtement cycliques adéquats. Nous formulons et démontrons ensuite une version orbifolde du théorème de semi-positivité générique de Y. Miyaoka : Ω 1 (X,Δ) est génériquement semi-positif si K X +Δ est pseudo-effectif. Nous en déduisons, à l’aide des résultats récents du PMML, un énoncé conjecturé par E. Viehweg : si X est lisse, et si Δ est un diviseur réduit à croisements normaux simples sur X tel qu’une puissance tensorielle de Ω X 1 (Log(Δ)) contienne un fibré en droites ‘big’, alors K X +Δ est lui-même ‘big’. Les travaux de Viehweg-Zuo impliquent alors la conjecture d’hyperbolicité de V.I. Shafarevich : si une famille algébrique de variétés projectives canoniquement polarisées et paramétrée par une variété quasi-projective irréductible lisse B a une ‘variation’ maximale, égale à dim(B), alors B est de type log-général.

Let X be a normal projective manifold, equipped with an effective ‘orbifold’ divisor Δ, such that the pair (X,Δ) is log-canonical. We first define the notion of ‘orbifold cotangent bundle’ Ω 1 (X,Δ), living on any suitable ramified cover of X. We are then in position to formulate and prove (in a completely different way) an orbifold version of Y. Miyaoka’s generic semi-positivity theorem: Ω 1 (X,Δ) is generically semi-positive if K X +Δ is pseudo-effective. Using the deep results of the LMMP, we immediately get a statement conjectured by E. Viehweg: if X is smooth, and if Δ is a reduced divisor with simple normal crossings on X such that some tensor power of Ω 1 (X,Δ)=Ω X 1 (Log(Δ)) contains the injective image of a big line bundle, then K X +Δ is big.

This implies, by fundamental results of Viehweg-Zuo, the ‘Shafarevich-Viehweg hyperbolicity conjecture’: if an algebraic family of canonically polarized manifolds parametrised by a quasi-projective manifold B has ‘maximal variation’, then B is of log-general type.

DOI : 10.5802/aif.2945
Classification : 14D05, 14D22, 14E22, 14E30, 14J40, 32J25
Keywords: Orbifold cotangent bundle, generic semi-positivity, canonically polarised manifolds
Mot clés : Fibré cotangent orbifolde, semi-positivité générique, variétés canoniquement polarisées
Campana, Frédéric 1 ; Păun, Mihai 2

1 Institut Elie Cartan Université Henri Poincaré B. P. 70239, F-54506 Vandoeuvre-lès-Nancy Cedex (France)
2 Korea Institute for Advanced Study School of Mathematics 85 Hoegiro, Dongdaemun-gu, Seoul 130-722 (Korea)
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Campana, Frédéric; Păun, Mihai. Orbifold generic semi-positivity: an application to families of canonically polarized manifolds. Annales de l'Institut Fourier, Tome 65 (2015) no. 2, pp. 835-861. doi : 10.5802/aif.2945. http://archive.numdam.org/articles/10.5802/aif.2945/

[1] Berndtsson, Bo; Păun, Mihai Quantitative extensions of pluricanonical forms and closed positive currents, Nagoya Math. J., Volume 205 (2012), pp. 25-65 http://projecteuclid.org/euclid.nmj/1330611001 | DOI | MR | Zbl

[2] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James Existence of minimal models for varieties of log general type (http://arxiv.org/abs/math/0610203) | MR | Zbl

[3] Bogomolov, Fedor; McQuillan, Michael Rational curves on foliated varieties (IHES preprint IHES/M/01/07.Février 2001.)

[4] Bost, Jean-Benoît Algebraic leaves of algebraic foliations over number fields, Publ. Math. Inst. Hautes Études Sci. (2001) no. 93, pp. 161-221 | DOI | Numdam | MR | Zbl

[5] Boucksom, Sébastien; Demailly, Jean-Pierre; Paun, Mihai; Peternell, Thomas The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension (http://arxiv.org/abs/math/0405285) | MR | Zbl

[6] Campana, Frédéric Special orbifolds and birational classification: a survey (http://arxiv.org/abs/1001.3763)

[7] Campana, Frédéric Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble), Volume 54 (2004) no. 3, pp. 499-630 | DOI | Numdam | MR | Zbl

[8] Campana, Frédéric Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes, J. Inst. Math. Jussieu, Volume 10 (2011) no. 4, pp. 809-934 | DOI | MR | Zbl

[9] Campana, Frédéric; Guenancia, Henri; Păun, Mihai Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields, Ann. Sci. Éc. Norm. Supér. (4), Volume 46 (2013) no. 6, pp. 879-916 | MR

[10] Campana, Frédéric; Păun, M. A differential-geometric approach for the generic semi-positivity of orbifold tensor bundles (in preparation)

[11] Campana, Frédéric; Peternell, Thomas Geometric stability of the cotangent bundle and the universal cover of a projective manifold, Bull. Soc. Math. France, Volume 139 (2011) no. 1, pp. 41-74 (With an appendix by Matei Toma) | Numdam | MR | Zbl

[12] Demailly, Jean-Pierre Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Algebraic geometry—Santa Cruz 1995 (Proc. Sympos. Pure Math.), Volume 62, Amer. Math. Soc., Providence, RI, 1997, pp. 285-360 | DOI | MR | Zbl

[13] Demailly, Jean-Pierre Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Pure Appl. Math. Q., Volume 7 (2011) no. 4, Special Issue: In memory of Eckart Viehweg, pp. 1165-1207 | DOI | MR

[14] Esnault, Hélène; Viehweg, Eckart Lectures on vanishing theorems, DMV Seminar, 20, Birkhäuser Verlag, Basel, 1992, pp. vi+164 | DOI | MR | Zbl

[15] Hartshorne, Robin Cohomological dimension of algebraic varieties, Ann. of Math. (2), Volume 88 (1968), pp. 403-450 | DOI | MR | Zbl

[16] Höring, A. On a conjecture of Beltrametti and Sommese (http://arxiv.org/abs/0912.1295) | Zbl

[17] Jabbusch, Kelly; Kebekus, Stefan Positive sheaves of differentials coming from coarse moduli spaces (http://arxiv.org/abs/0904.2445) | Numdam | MR | Zbl

[18] Jabbusch, Kelly; Kebekus, Stefan Families over special base manifolds and a conjecture of Campana, Math. Z., Volume 269 (2011) no. 3-4, pp. 847-878 | DOI | MR | Zbl

[19] Kawamata, Yujiro Subadjunction of log canonical divisors for a subvariety of codimension 2, Birational algebraic geometry (Baltimore, MD, 1996) (Contemp. Math.), Volume 207, Amer. Math. Soc., Providence, RI, 1997, pp. 79-88 | DOI | MR | Zbl

[20] Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985 (Adv. Stud. Pure Math.), Volume 10, North-Holland, Amsterdam, 1987, pp. 283-360 | MR | Zbl

[21] Kebekus, Stefan Differential forms on singular spaces, the minimal program, and hyperbolicity of moduli (http://arxiv.org/abs/1107.4239)

[22] Kebekus, Stefan; Kovács, Sándor J. Families of canonically polarized varieties over surfaces, Invent. Math., Volume 172 (2008) no. 3, pp. 657-682 | DOI | MR | Zbl

[23] Kebekus, Stefan; Kovács, Sándor J. The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties, Duke Math. J., Volume 155 (2010) no. 1, pp. 1-33 | DOI | MR | Zbl

[24] Kebekus, Stefan; Solá Conde, Luis; Toma, Matei Rationally connected foliations after Bogomolov and McQuillan, J. Algebraic Geom., Volume 16 (2007) no. 1, pp. 65-81 | DOI | MR | Zbl

[25] Kollár, János Lectures on resolution of singularities, Annals of Mathematics Studies, 166, Princeton University Press, Princeton, NJ, 2007, pp. vi+208 | MR | Zbl

[26] Langer, A. Logarithmic orbifold Euler numbers of surfaces with applications (http://arxiv.org/abs/math/0012180) | MR | Zbl

[27] Miyaoka, Yoichi Deformations of a morphism along a foliation and applications, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) (Proc. Sympos. Pure Math.), Volume 46, Amer. Math. Soc., Providence, RI, 1987, pp. 245-268 | MR | Zbl

[28] Paršin, A. N. Algebraic curves over function fields, Dokl. Akad. Nauk SSSR, Volume 183 (1968), pp. 524-526 | MR | Zbl

[29] Patakfalvi, Zsolt Viehweg hyperbolicity conjecture is true over compact bases (http://arxiv.org/abs/1109.2835) | MR | Zbl

[30] Taji, Behrouz The isotriviality of families of canonically-polarised manifolds over a special quasi-projective base (http://arxiv.org/abs/1310.5391)

[31] Viehweg, Eckart; Zuo, Kang Base spaces of non-isotrivial families of smooth minimal models, Complex geometry (Göttingen, 2000), Springer, Berlin, 2002, pp. 279-328 | MR | Zbl

[32] Yano, K.; Bochner, S. Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953, pp. ix+190 | MR | Zbl

[33] Yau, Shing Tung On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math., Volume 31 (1978) no. 3, pp. 339-411 | DOI | MR | Zbl

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