On the positivity of the logarithmic cotangent bundle

Brotbek, Damian; Deng, Ya

Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 3001-3051.

L’objectif de ce travail est de construire des exemples de paires dont le fibré cotangent logarithmique possède de fortes propriétés de positivité. Ces exemples sont construit à partir de n’importe quelle variété lisse de dimension $n$ en considérant la somme d’au moins $n$ diviseurs généraux suffisamment amples.

The aim of this work is to construct examples of pairs whose logarithmic cotangent bundles have strong positivity properties. These examples are constructed from any smooth $n$-dimensional complex projective varieties by considering the sum of at least $n$ general sufficiently ample hypersurfaces.

Classification : 14J60, 32Q45
Keywords: Logarithmic cotangent bundles, hyperbolicity
Mot clés : Fibré cotangent logarithmique, hyperbolicité

[1] Benoist, Olivier Le théorème de Bertini en famille, Bull. Soc. Math. Fr., Volume 139 (2011) no. 4, pp. 555-569 | Zbl

[2] Berndtsson, Bo; Păun, Mihai; Wang, Xu Algebraic fiber spaces and curvature of higher direct images (2017) (https://arxiv.org/abs/1704.02279)

[3] Brotbek, Damian Differential equations as embedding obstructions and vanishing theorems (2011) (https://arxiv.org/abs/1111.5324)

[4] Brotbek, Damian Symmetric differential forms on complete intersection varieties and applications, Math. Ann., Volume 366 (2016) no. 1-2, pp. 417-446 | DOI | MR

[5] Brotbek, Damian On the hyperbolicity of general hypersurfaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 126 (2017), pp. 1-34 | DOI | MR

[6] Brotbek, Damian; Darondeau, Lionel Complete intersection varieties with ample cotangent bundles, Invent. Math., Volume 212 (2018) no. 3, pp. 913-940 | DOI | MR

[7] Brotbek, Damian; Deng, Ya Hyperbolicity of the complements of general hypersurfaces of high degree (2018) (https://arxiv.org/abs/1804.01719)

[8] Brückmann, Peter; Rackwitz, Hans-Georg $T$-symmetrical tensor forms on complete intersections, Math. Ann., Volume 288 (1990) no. 4, pp. 627-635 | DOI | MR | Zbl

[9] Brunebarbe, Yohan Symmetric differentials and variations of Hodge structures, J. Reine Angew. Math., Volume 743 (2018), pp. 133-161 | Zbl

[10] Brunebarbe, Yohan; Cadorel, Benoit Hyperbolicity of varieties supporting a variation of Hodge structure (2017) (https://arxiv.org/abs/1707.01327)

[11] Cadorel, Benoit Symmetric differentials on complex hyperbolic manifolds with cusps (2016) (https://arxiv.org/abs/1606.05470)

[12] Darondeau, Lionel On the logarithmic Green-Griffiths conjecture, Int. Math. Res. Not. (2016) no. 6, pp. 1871-1923 | DOI | MR

[13] Debarre, Olivier Varieties with ample cotangent bundle, Compos. Math., Volume 141 (2005) no. 6, pp. 1445-1459 | DOI

[14] Demailly, Jean-Pierre Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Algebraic geometry (Santa Cruz, 1995) (Proceedings of Symposia in Pure Mathematics), Volume 62, American Mathematical Society, 1997, pp. 285-360 | MR | Zbl

[15] Deng, Ya Effectivity in the Hyperbolicity-related problems (2016) (https://arxiv.org/abs/1606.03831)

[16] Deng, Ya On the Diverio-Trapani Conjecture (2017) (https://arxiv.org/abs/1703.07560, to appear in Ann. Sci. Éc. Norm. Supér.)

[17] Deng, Ya Kobayashi hyperbolicity of moduli spaces of minimal projective manifolds of general type (with the appendix by Dan Abramovich) (2018) (https://arxiv.org/abs/1806.01666)

[18] Deng, Ya Pseudo Kobayashi hyperbolicity of base spaces of families of minimal projective manifolds with maximal variation (2018) (https://arxiv.org/abs/1809.05891)

[19] Diverio, Simone Existence of global invariant jet differentials on projective hypersurfaces of high degree, Math. Ann., Volume 344 (2009) no. 2, pp. 293-315 | DOI | MR

[20] Diverio, Simone; Merker, Joël; Rousseau, Erwan Effective algebraic degeneracy, Invent. Math., Volume 180 (2010) no. 1, pp. 161-223 | DOI | MR

[21] El Goul, Jawher Logarithmic jets and hyperbolicity, Osaka J. Math., Volume 40 (2003) no. 2, pp. 469-491 | MR | Zbl

[22] Green, Mark L. The hyperbolicity of the complement of $2n+1$ hyperplanes in general position in $P_n$ and related results, Proc. Am. Math. Soc., Volume 66 (1977) no. 1, pp. 109-113 | DOI | MR

[23] Kobayashi, Shoshichi Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften, 318, Springer, 1998, xiv+471 pages | DOI | MR

[24] Lazarsfeld, Robert Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 49, Springer, 2004, xviii+385 pages (Positivity for vector bundles, and multiplier ideals)

[25] Lu, Steven S. Y.; Winkelmann, Jörg Quasiprojective varieties admitting Zariski dense entire holomorphic curves, Forum Math., Volume 24 (2012) no. 2, pp. 399-418 | DOI | MR | Zbl

[26] McQuillan, Michael Diophantine approximations and foliations, Publ. Math., Inst. Hautes Étud. Sci. (1998) no. 87, pp. 121-174 | MR

[27] Nakamaye, Michael Stable base loci of linear series, Math. Ann., Volume 318 (2000) no. 4, pp. 837-847 | DOI

[28] Noguchi, Junjiro Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties, Nagoya Math. J., Volume 83 (1981), pp. 213-233 http://projecteuclid.org.scd-rproxy.u-strasbg.fr/euclid.nmj/1118786486 | MR

[29] Noguchi, Junjiro Logarithmic jet spaces and extensions of de Franchis’ theorem, Contributions to several complex variables (Aspects of Mathematics), Volume E9, Vieweg & Sohn, 1986, pp. 227-249 | MR | Zbl

[30] Noguchi, Junjiro; Winkelmann, Jörg Nevanlinna theory in several complex variables and Diophantine approximation, Grundlehren der Mathematischen Wissenschaften, 350, Springer, 2014, xiv+416 pages | DOI | MR

[31] Noguchi, Junjiro; Winkelmann, Jörg; Yamanoi, Katsutoshi Degeneracy of holomorphic curves into algebraic varieties, J. Math. Pures Appl., Volume 88 (2007) no. 3, pp. 293-306 | DOI | MR | Zbl

[32] Noguchi, Junjiro; Winkelmann, Jörg; Yamanoi, Katsutoshi The second main theorem for holomorphic curves into semi-abelian varieties. II, Forum Math., Volume 20 (2008) no. 3, pp. 469-503 | DOI | MR

[33] Noguchi, Junjiro; Winkelmann, Jörg; Yamanoi, Katsutoshi Degeneracy of holomorphic curves into algebraic varieties II, Vietnam J. Math., Volume 41 (2013) no. 4, pp. 519-525 | DOI | MR | Zbl

[34] Rousseau, Erwan Hyperbolicité du complémentaire d’une courbe dans ${\mathbb{P}}^2$: le cas de deux composantes, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 8, pp. 635-640 | DOI | MR

[35] Sakai, Fumio Symmetric powers of the cotangent bundle and classification of algebraic varieties, Algebraic geometry (Copenhagen, 1978) (Lecture Notes in Mathematics), Volume 732, Springer, 1979, pp. 545-563 | MR | Zbl

[36] Schneider, Michael Symmetric differential forms as embedding obstructions and vanishing theorems, J. Algebr. Geom., Volume 1 (1992) no. 2, pp. 175-181 | Zbl

[37] Siu, Yum-Tong Hyperbolicity in complex geometry, The legacy of Niels Henrik Abel (Oslo, 2012), Springer, 2004, pp. 543-566 | MR | Zbl

[38] Siu, Yum-Tong Hyperbolicity of generic high-degree hypersurfaces in complex projective space, Invent. Math., Volume 202 (2015) no. 3, pp. 1069-1166 | DOI | MR

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

[40] Voisin, Claire On a conjecture of Clemens on rational curves on hypersurfaces, J. Differ. Geom., Volume 44 (1996) no. 1, pp. 200-213 | MR | Zbl

[41] Voisin, Claire A correction: “On a conjecture of Clemens on rational curves on hypersurfaces” [J. Differential Geom. 44 (1996), no. 1, 200–213], J. Differ. Geom., Volume 49 (1998) no. 3, pp. 601-611 | MR | Zbl

[42] Xie, Song-Yan On the ampleness of the cotangent bundles of complete intersections, Invent. Math., Volume 212 (2018) no. 3, pp. 941-996 | DOI | MR

[43] Zuo, Kang On the negativity of kernels of Kodaira-Spencer maps on Hodge bundles and applications, Asian J. Math., Volume 4 (2000) no. 1, pp. 279-301 | DOI | MR