In this paper we study the positivity of the cotangent bundle of projective manifolds. We conjecture that the cotangent bundle is pseudoeffective if and only the manifold has non-zero symmetric differentials. We confirm this conjecture for most projective surfaces that are not of general type.
Dans ce papier nous étudions la positivité du fibré cotangent des variétés projective lisses. Nous conjecturons que le fibré cotangent est pseudoeffectif si et seulement si la variété possède des formes holomorphes symétriques non-nulles. Nous montrons cette conjecture pour la plupart des surfaces projectives lisses qui ne sont pas de type général.
Accepted:
Published online:
Keywords: MMP, minimal models, positivity of vector bundles, nonvanishing conjecture, symmetric differentials
Andreas Höring 1; Thomas Peternell 2
CC-BY 4.0
@article{AFST_2023_6_32_5_855_0,
author = {Andreas H\"oring and Thomas Peternell},
title = {A {Nonvanishing} {Conjecture} for {Cotangent} {Bundles}},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {855--892},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 32},
number = {5},
year = {2023},
doi = {10.5802/afst.1756},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1756/}
}
TY - JOUR AU - Andreas Höring AU - Thomas Peternell TI - A Nonvanishing Conjecture for Cotangent Bundles JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2023 SP - 855 EP - 892 VL - 32 IS - 5 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1756/ DO - 10.5802/afst.1756 LA - en ID - AFST_2023_6_32_5_855_0 ER -
%0 Journal Article %A Andreas Höring %A Thomas Peternell %T A Nonvanishing Conjecture for Cotangent Bundles %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2023 %P 855-892 %V 32 %N 5 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1756/ %R 10.5802/afst.1756 %G en %F AFST_2023_6_32_5_855_0
Andreas Höring; Thomas Peternell. A Nonvanishing Conjecture for Cotangent Bundles. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 5, pp. 855-892. doi : 10.5802/afst.1756. https://afst.centre-mersenne.org/articles/10.5802/afst.1756/
[1] Modifications analytiques, Lecture Notes in Mathematics, 943, Springer, 1982, iv+120 pages | DOI | MR
[2] A view on contractions of higher-dimensional varieties, Algebraic geometry—Santa Cruz 1995 (Proceedings of Symposia in Pure Mathematics), Volume 62, American Mathematical Society, 1997, pp. 153-183 | DOI | MR | Zbl
[3] Rational curves on fibered varieties, Math. Z., Volume 298 (2021) no. 3-4, pp. 1097-1111 | DOI | MR | Zbl
[4] Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 4, Springer, 2004, xii+436 pages | DOI | MR
[5] A simple proof for the existence of Zariski decompositions on surfaces, J. Algebr. Geom., Volume 18 (2009) no. 4, pp. 789-793 | DOI | MR
[6] Géométrie de la projectivisation des idéaux et applications aux problèmes de birationalité, Ph. D. Thesis, Institut de Mathématiques de Bourgogne (Dijon) (2018)
[7] Existence of minimal models for varieties of log general type, J. Am. Math. Soc., Volume 23 (2010) no. 2, pp. 405-468 | DOI | MR | Zbl
[8] Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 1, pp. 45-76 | DOI | Numdam | MR | Zbl
[9] The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebr. Geom., Volume 22 (2013), pp. 201-248 | DOI
[10] Hyperbolicity related problems for complete intersection varieties, Compos. Math., Volume 150 (2014) no. 3, pp. 369-395 | DOI | MR | Zbl
[11] Explicit computation of symmetric differentials and its application to quasi-hyperbolicity, Algebra Number Theory, Volume 16 (2022) no. 6, pp. 1377-1405 | DOI
[12] Symmetric differentials and the fundamental group, Duke Math. J., Volume 162 (2013) no. 14, pp. 2797-2813 | DOI | MR
[13] A positivity property for foliations on compact Kähler manifolds, Int. J. Math., Volume 17 (2006) no. 1, pp. 35-43 | DOI | MR
[14] Negativity of compact curves in infinite covers of projective surfaces, J. Algebr. Geom., Volume 7 (1998) no. 4, pp. 673-693 | MR
[15] Orbifolds, special varieties and classification theory, Ann. Inst. Fourier, Volume 54 (2004) no. 3, pp. 499-630 | DOI | Numdam | MR
[16] 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
[17] Rationally connected manifolds and semipositivity of the Ricci curvature, Recent advances in algebraic geometry (London Mathematical Society Lecture Note Series), Volume 417, Cambridge University Press, 2015, pp. 71-91 | DOI | MR | Zbl
[18] Intersection numbers of sections of elliptic surfaces, Invent. Math., Volume 53 (1979) no. 1, pp. 1-44 | DOI | MR
[19] Higher-dimensional algebraic geometry, Universitext, Springer, 2001, xiv+233 pages | MR
[20] Compact complex manifolds with numerically effective tangent bundles, J. Algebr. Geom., Volume 3 (1994) no. 2, pp. 295-345 | MR | Zbl
[21] A decomposition theorem for singular spaces with trivial canonical class of dimension at most five, Invent. Math., Volume 211 (2018) no. 1, pp. 245-296 | DOI | MR | Zbl
[22] Riemann surfaces, Graduate Texts in Mathematics, 71, Springer, 1980, xi+337 pages | DOI | MR
[23] Positivity of the cotangent sheaf of singular Calabi–Yau varieties, Math. Res. Lett. (2022) no. 2, pp. 339-371 | DOI | MR
[24] Families of rationally connected varieties, J. Am. Math. Soc., Volume 16 (2003) no. 1, p. 57-67 (electronic) | DOI | MR
[25] Differential forms on log canonical spaces, Publ. Math., Inst. Hautes Étud. Sci. (2011) no. 114, pp. 87-169 | DOI | Numdam | MR
[26] Etale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties, Duke Math. J., Volume 165 (2016) no. 10, pp. 1965-2004 | DOI | MR
[27] Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977, xvi+496 pages | DOI | MR
[28] Examples of Fano manifolds with non-pseudoeffective tangent bundle, J. Lond. Math. Soc., Volume 106 (2022) no. 1, pp. 27-59 | DOI | MR
[29] Algebraic integrability of foliations with numerically trivial canonical bundle, Invent. Math., Volume 216 (2019) no. 2, pp. 395-419 | DOI | MR
[30] Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 32, Springer, 1996, viii+320 pages | DOI | MR
[31] Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, 1998, viii+254 pages (with the collaboration of C. H. Clemens and A. Corti) | DOI | MR
[32] Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 48, Springer, 2004, xviii+387 pages | MR
[33] Sur les algèbres universelles, Ann. Inst. Fourier, Volume 14 (1964) no. 2, pp. 33-87 | DOI | Numdam | MR
[34] The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica, ETS Editrice, 1989, vi+108 pages | MR
[35] Zariski-decomposition and abundance, MSJ Memoirs, 14, Mathematical Society of Japan, 2004, xiv+277 pages | MR
[36] Symmetric powers of the cotangent bundle and classification of algebraic varieties, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978) (Lecture Notes in Mathematics), Volume 732, Springer, 1979, pp. 545-563 | DOI | MR | Zbl
[37] Isotrivial fibred surfaces, Ann. Mat. Pura Appl., Volume 171 (1996), pp. 63-81 | DOI | MR | Zbl
[38] Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, 439, Springer, 1975, xix+278 pages (notes written in collaboration with P. Cherenack) | DOI | MR
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