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 -dimensional complex projective varieties by considering the sum of at least general sufficiently ample hypersurfaces.
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 en considérant la somme d’au moins diviseurs généraux suffisamment amples.
Keywords: Logarithmic cotangent bundles, hyperbolicity
Mot clés : Fibré cotangent logarithmique, hyperbolicité
@article{AIF_2018__68_7_3001_0, author = {Brotbek, Damian and Deng, Ya}, title = {On the positivity of the logarithmic cotangent bundle}, journal = {Annales de l'Institut Fourier}, pages = {3001--3051}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {7}, year = {2018}, doi = {10.5802/aif.3235}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.3235/} }
TY - JOUR AU - Brotbek, Damian AU - Deng, Ya TI - On the positivity of the logarithmic cotangent bundle JO - Annales de l'Institut Fourier PY - 2018 SP - 3001 EP - 3051 VL - 68 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.3235/ DO - 10.5802/aif.3235 LA - en ID - AIF_2018__68_7_3001_0 ER -
%0 Journal Article %A Brotbek, Damian %A Deng, Ya %T On the positivity of the logarithmic cotangent bundle %J Annales de l'Institut Fourier %D 2018 %P 3001-3051 %V 68 %N 7 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.3235/ %R 10.5802/aif.3235 %G en %F AIF_2018__68_7_3001_0
Brotbek, Damian; Deng, Ya. On the positivity of the logarithmic cotangent bundle. Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 3001-3051. doi : 10.5802/aif.3235. http://www.numdam.org/articles/10.5802/aif.3235/
[1] Le théorème de Bertini en famille, Bull. Soc. Math. Fr., Volume 139 (2011) no. 4, pp. 555-569 | Zbl
[2] Algebraic fiber spaces and curvature of higher direct images (2017) (https://arxiv.org/abs/1704.02279)
[3] Differential equations as embedding obstructions and vanishing theorems (2011) (https://arxiv.org/abs/1111.5324)
[4] Symmetric differential forms on complete intersection varieties and applications, Math. Ann., Volume 366 (2016) no. 1-2, pp. 417-446 | DOI | MR
[5] On the hyperbolicity of general hypersurfaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 126 (2017), pp. 1-34 | DOI | MR
[6] Complete intersection varieties with ample cotangent bundles, Invent. Math., Volume 212 (2018) no. 3, pp. 913-940 | DOI | MR
[7] Hyperbolicity of the complements of general hypersurfaces of high degree (2018) (https://arxiv.org/abs/1804.01719)
[8] -symmetrical tensor forms on complete intersections, Math. Ann., Volume 288 (1990) no. 4, pp. 627-635 | DOI | MR | Zbl
[9] Symmetric differentials and variations of Hodge structures, J. Reine Angew. Math., Volume 743 (2018), pp. 133-161 | Zbl
[10] Hyperbolicity of varieties supporting a variation of Hodge structure (2017) (https://arxiv.org/abs/1707.01327)
[11] Symmetric differentials on complex hyperbolic manifolds with cusps (2016) (https://arxiv.org/abs/1606.05470)
[12] On the logarithmic Green-Griffiths conjecture, Int. Math. Res. Not. (2016) no. 6, pp. 1871-1923 | DOI | MR
[13] Varieties with ample cotangent bundle, Compos. Math., Volume 141 (2005) no. 6, pp. 1445-1459 | DOI
[14] 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] Effectivity in the Hyperbolicity-related problems (2016) (https://arxiv.org/abs/1606.03831)
[16] On the Diverio-Trapani Conjecture (2017) (https://arxiv.org/abs/1703.07560, to appear in Ann. Sci. Éc. Norm. Supér.)
[17] 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] Pseudo Kobayashi hyperbolicity of base spaces of families of minimal projective manifolds with maximal variation (2018) (https://arxiv.org/abs/1809.05891)
[19] 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] Effective algebraic degeneracy, Invent. Math., Volume 180 (2010) no. 1, pp. 161-223 | DOI | MR
[21] Logarithmic jets and hyperbolicity, Osaka J. Math., Volume 40 (2003) no. 2, pp. 469-491 | MR | Zbl
[22] The hyperbolicity of the complement of hyperplanes in general position in and related results, Proc. Am. Math. Soc., Volume 66 (1977) no. 1, pp. 109-113 | DOI | MR
[23] Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften, 318, Springer, 1998, xiv+471 pages | DOI | MR
[24] 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] Quasiprojective varieties admitting Zariski dense entire holomorphic curves, Forum Math., Volume 24 (2012) no. 2, pp. 399-418 | DOI | MR | Zbl
[26] Diophantine approximations and foliations, Publ. Math., Inst. Hautes Étud. Sci. (1998) no. 87, pp. 121-174 | MR
[27] Stable base loci of linear series, Math. Ann., Volume 318 (2000) no. 4, pp. 837-847 | DOI
[28] 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] 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] Nevanlinna theory in several complex variables and Diophantine approximation, Grundlehren der Mathematischen Wissenschaften, 350, Springer, 2014, xiv+416 pages | DOI | MR
[31] Degeneracy of holomorphic curves into algebraic varieties, J. Math. Pures Appl., Volume 88 (2007) no. 3, pp. 293-306 | DOI | MR | Zbl
[32] 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] Degeneracy of holomorphic curves into algebraic varieties II, Vietnam J. Math., Volume 41 (2013) no. 4, pp. 519-525 | DOI | MR | Zbl
[34] Hyperbolicité du complémentaire d’une courbe dans : le cas de deux composantes, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 8, pp. 635-640 | DOI | MR
[35] 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] Symmetric differential forms as embedding obstructions and vanishing theorems, J. Algebr. Geom., Volume 1 (1992) no. 2, pp. 175-181 | Zbl
[37] Hyperbolicity in complex geometry, The legacy of Niels Henrik Abel (Oslo, 2012), Springer, 2004, pp. 543-566 | MR | Zbl
[38] Hyperbolicity of generic high-degree hypersurfaces in complex projective space, Invent. Math., Volume 202 (2015) no. 3, pp. 1069-1166 | DOI | MR
[39] Base spaces of non-isotrivial families of smooth minimal models, Complex geometry (Göttingen, 2000), Springer, 2002, pp. 279-328 | MR | Zbl
[40] On a conjecture of Clemens on rational curves on hypersurfaces, J. Differ. Geom., Volume 44 (1996) no. 1, pp. 200-213 | MR | Zbl
[41] 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] On the ampleness of the cotangent bundles of complete intersections, Invent. Math., Volume 212 (2018) no. 3, pp. 941-996 | DOI | MR
[43] 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
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