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.
Classification: 14J60, 32Q45
Keywords: Logarithmic cotangent bundles, hyperbolicity
@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 = {https://aif.centre-mersenne.org/articles/10.5802/aif.3235/} }
TY - JOUR TI - On the positivity of the logarithmic cotangent bundle JO - Annales de l'Institut Fourier PY - 2018 DA - 2018/// SP - 3001 EP - 3051 VL - 68 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3235/ UR - https://doi.org/10.5802/aif.3235 DO - 10.5802/aif.3235 LA - en ID - AIF_2018__68_7_3001_0 ER -
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. https://aif.centre-mersenne.org/articles/10.5802/aif.3235/
[1] Le théorème de Bertini en famille, Bull. Soc. Math. Fr., Tome 139 (2011) no. 4, pp. 555-569 | Zbl: 1244.14045
[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., Tome 366 (2016) no. 1-2, pp. 417-446 | Article | MR: 3552245
[5] On the hyperbolicity of general hypersurfaces, Publ. Math., Inst. Hautes Étud. Sci., Tome 126 (2017), pp. 1-34 | Article | MR: 3735863
[6] Complete intersection varieties with ample cotangent bundles, Invent. Math., Tome 212 (2018) no. 3, pp. 913-940 | Article | MR: 3802300
[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., Tome 288 (1990) no. 4, pp. 627-635 | Article | MR: 1081268 | Zbl: 0724.14032
[9] Symmetric differentials and variations of Hodge structures, J. Reine Angew. Math., Tome 743 (2018), pp. 133-161 | Zbl: 06950104
[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 | Article | MR: 3509943
[13] Varieties with ample cotangent bundle, Compos. Math., Tome 141 (2005) no. 6, pp. 1445-1459 | Article
[14] Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Algebraic geometry (Santa Cruz, 1995) (Proceedings of Symposia in Pure Mathematics) Tome 62, American Mathematical Society, 1997, pp. 285-360 | MR: 1492539 | Zbl: 0919.32014
[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., Tome 344 (2009) no. 2, pp. 293-315 | Article | MR: 2495771
[20] Effective algebraic degeneracy, Invent. Math., Tome 180 (2010) no. 1, pp. 161-223 | Article | MR: 2593279
[21] Logarithmic jets and hyperbolicity, Osaka J. Math., Tome 40 (2003) no. 2, pp. 469-491 | MR: 1988702 | Zbl: 1048.32016
[22] The hyperbolicity of the complement of hyperplanes in general position in and related results, Proc. Am. Math. Soc., Tome 66 (1977) no. 1, pp. 109-113 | Article | MR: 0457790
[23] Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften, Tome 318, Springer, 1998, xiv+471 pages | Article | MR: 1635983
[24] Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Tome 49, Springer, 2004, xviii+385 pages (Positivity for vector bundles, and multiplier ideals)
[25] Quasiprojective varieties admitting Zariski dense entire holomorphic curves, Forum Math., Tome 24 (2012) no. 2, pp. 399-418 | Article | MR: 2900013 | Zbl: 1273.32023
[26] Diophantine approximations and foliations, Publ. Math., Inst. Hautes Étud. Sci. (1998) no. 87, pp. 121-174 | MR: 1659270
[27] Stable base loci of linear series, Math. Ann., Tome 318 (2000) no. 4, pp. 837-847 | Article
[28] Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties, Nagoya Math. J., Tome 83 (1981), pp. 213-233 http://projecteuclid.org.scd-rproxy.u-strasbg.fr/euclid.nmj/1118786486 | MR: 632655
[29] Logarithmic jet spaces and extensions of de Franchis’ theorem, Contributions to several complex variables (Aspects of Mathematics) Tome E9, Vieweg & Sohn, 1986, pp. 227-249 | MR: 859200 | Zbl: 0598.32021
[30] Nevanlinna theory in several complex variables and Diophantine approximation, Grundlehren der Mathematischen Wissenschaften, Tome 350, Springer, 2014, xiv+416 pages | Article | MR: 3156076
[31] Degeneracy of holomorphic curves into algebraic varieties, J. Math. Pures Appl., Tome 88 (2007) no. 3, pp. 293-306 | Article | MR: 2355461 | Zbl: 1135.32018
[32] The second main theorem for holomorphic curves into semi-abelian varieties. II, Forum Math., Tome 20 (2008) no. 3, pp. 469-503 | Article | MR: 2418202
[33] Degeneracy of holomorphic curves into algebraic varieties II, Vietnam J. Math., Tome 41 (2013) no. 4, pp. 519-525 | Article | MR: 3142409 | Zbl: 1293.32022
[34] Hyperbolicité du complémentaire d’une courbe dans : le cas de deux composantes, C. R. Math. Acad. Sci. Paris, Tome 336 (2003) no. 8, pp. 635-640 | Article | MR: 1988123
[35] Symmetric powers of the cotangent bundle and classification of algebraic varieties, Algebraic geometry (Copenhagen, 1978) (Lecture Notes in Mathematics) Tome 732, Springer, 1979, pp. 545-563 | MR: 555717 | Zbl: 0415.14020
[36] Symmetric differential forms as embedding obstructions and vanishing theorems, J. Algebr. Geom., Tome 1 (1992) no. 2, pp. 175-181 | Zbl: 0790.14009
[37] Hyperbolicity in complex geometry, The legacy of Niels Henrik Abel (Oslo, 2012), Springer, 2004, pp. 543-566 | MR: 2077584 | Zbl: 10796.32011
[38] Hyperbolicity of generic high-degree hypersurfaces in complex projective space, Invent. Math., Tome 202 (2015) no. 3, pp. 1069-1166 | Article | MR: 3425387
[39] Base spaces of non-isotrivial families of smooth minimal models, Complex geometry (Göttingen, 2000), Springer, 2002, pp. 279-328 | MR: 1922109 | Zbl: 1006.14004
[40] On a conjecture of Clemens on rational curves on hypersurfaces, J. Differ. Geom., Tome 44 (1996) no. 1, pp. 200-213 | MR: 1420353 | Zbl: 0883.14022
[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., Tome 49 (1998) no. 3, pp. 601-611 | MR: 1669712 | Zbl: 0994.14026
[42] On the ampleness of the cotangent bundles of complete intersections, Invent. Math., Tome 212 (2018) no. 3, pp. 941-996 | Article | MR: 3802301
[43] On the negativity of kernels of Kodaira-Spencer maps on Hodge bundles and applications, Asian J. Math., Tome 4 (2000) no. 1, pp. 279-301 | Article | MR: 1803724
Cited by Sources: