A decomposition theorem for projective manifolds with nef anticanonical bundle
Authors:
Junyan Cao and Andreas Höring
Journal:
J. Algebraic Geom. 28 (2019), 567-597
DOI:
https://doi.org/10.1090/jag/715
Published electronically:
April 25, 2019
MathSciNet review:
3959071
Full-text PDF
Abstract |
References |
Additional Information
Abstract: Let $X$ be a simply connected projective manifold with nef anticanonical bundle. We prove that $X$ is a product of a rationally connected manifold and a manifold with trivial canonical bundle. As an application we describe the MRC-fibration of any projective manifold with nef anticanonical bundle.
References
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- Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472
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- Mihai Paun, Sur le groupe fondamental des variétés kählériennes compactes à classe de Ricci numériquement effective, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 11, 1249–1254 (French, with English and French summaries). MR 1456296, DOI https://doi.org/10.1016/S0764-4442%2899%2980408-X
- Mihai Păun, Singular Hermitian metrics and positivity of direct images of pluricanonical bundles, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 519–553. MR 3821161
- Mihai Păun and Shigeharu Takayama, Positivity of twisted relative pluricanonical bundles and their direct images, J. Algebraic Geom. 27 (2018), no. 2, 211–272. MR 3764276, DOI https://doi.org/10.1090/S1056-3911-2017-00702-1
- Hossein Raufi, Singular hermitian metrics on holomorphic vector bundles, Ark. Mat. 53 (2015), no. 2, 359–382. MR 3391176, DOI https://doi.org/10.1007/s11512-015-0212-4
- Miles Reid, Nonnormal del Pezzo surfaces, Publ. Res. Inst. Math. Sci. 30 (1994), no. 5, 695–727. MR 1311389, DOI https://doi.org/10.2977/prims/1195165581
- Eckart Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30, Springer-Verlag, Berlin, 1995. MR 1368632
- Qi Zhang, On projective varieties with nef anticanonical divisors, Math. Ann. 332 (2005), no. 3, 697–703. MR 2181470, DOI https://doi.org/10.1007/s00208-005-0649-z
References
- Arnaud Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755–782 (1984) (French). MR 730926
- Bo Berndtsson, Curvature of vector bundles associated to holomorphic fibrations, Ann. of Math. (2) 169 (2009), no. 2, 531–560. MR 2480611
- Thomas Bauer and Thomas Peternell, Nef reduction and anticanonical bundles, Asian J. Math. 8 (2004), no. 2, 315–352. MR 2129540
- Bo Berndtsson and Mihai Păun, Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J. 145 (2008), no. 2, 341–378. MR 2449950
- Frédéric Campana, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, 499–630 (English, with English and French summaries). MR 2097416
- Junyan Cao, Albanese maps of projective manifolds with nef anticanonical bundles, arXiv:1612.05921, 2016.
- F. Campana, J.-P. Demailly, and Th. Peternell, Rationally connected manifolds and semipositivity of the Ricci curvature, Recent advances in algebraic geometry, London Math. Soc. Lecture Note Ser., vol. 417, Cambridge Univ. Press, Cambridge, 2015, pp. 71–91. MR 3380444
- Junyan Cao and Andreas Höring, Manifolds with nef anticanonical bundle, J. Reine Angew. Math. 724 (2017), 203–244. MR 3619107
- César Camacho and Alcides Lins Neto, Geometric theory of foliations, Birkhäuser Boston, Inc., Boston, MA, 1985. Translated from the Portuguese by Sue E. Goodman. MR 824240
- Frédéric Campana and Thomas Peternell, Projective manifolds whose tangent bundles are numerically effective, Math. Ann. 289 (1991), no. 1, 169–187. MR 1087244
- Junyan Cao and Mihai Păun, Kodaira dimension of algebraic fiber spaces over abelian varieties, Invent. Math. 207 (2017), no. 1, 345–387. MR 3592759
- Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091
- Jean-Pierre Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), no. 3, 361–409. MR 1158622
- Jean-Pierre Demailly, Analytic methods in algebraic geometry, Surveys of Modern Mathematics, vol. 1, International Press, Somerville, MA; Higher Education Press, Beijing, 2012. MR 2978333
- Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Kähler manifolds with numerically effective Ricci class, Compositio Math. 89 (1993), no. 2, 217–240. MR 1255695
- Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), no. 2, 295–345. MR 1257325
- Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Compact Kähler manifolds with Hermitian semipositive anticanonical bundle, Compositio Math. 101 (1996), no. 2, 217–224. MR 1389367
- Wolfgang Fischer and Hans Grauert, Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1965 (1965), 89–94 (German). MR 0184258
- Takao Fujita, On Kähler fiber spaces over curves, J. Math. Soc. Japan 30 (1978), no. 4, 779–794. MR 513085
- Tom Graber, Joe Harris, and Jason Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), no. 1, 57–67. MR 1937199
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Robin Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121–176. MR 597077
- Andreas Höring, Uniruled varieties with split tangent bundle, Math. Z. 256 (2007), no. 3, 465–479. MR 2299565
- Christopher Hacon, Mihnea Popa, and Christian Schnell, Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Păun, Local and global methods in algebraic geometry, Contemp. Math., vol. 712, Amer. Math. Soc., Providence, RI, 2018, pp. 143–195. MR 3832403, DOI https://doi.org/10.1090/conm/712/14346
- Ludger Kaup and Burchard Kaup, Holomorphic functions of several variables: An introduction to the fundamental theory, with the assistance of Gottfried Barthel, translated from the German by Michael Bridgland, De Gruyter Studies in Mathematics, vol. 3, Walter de Gruyter & Co., Berlin, 1983. MR 716497
- Robert Lazarsfeld, Positivity in algebraic geometry. I: Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. MR 2095471
- Robert Lazarsfeld, Positivity in algebraic geometry. II: Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. MR 2095472
- Steven Lu, Yuping Tu, Qi Zhang, and Quan Zheng, On semistability of Albanese maps, Manuscripta Math. 131 (2010), no. 3-4, 531–535. MR 2592095
- Mihai Paun, Sur le groupe fondamental des variétés kählériennes compactes à classe de Ricci numériquement effective, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 11, 1249–1254 (French, with English and French summaries). MR 1456296
- Mihai Păun, Singular Hermitian metrics and positivity of direct images of pluricanonical bundles, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 519–553. MR 3821161
- Mihai Păun and Shigeharu Takayama, Positivity of twisted relative pluricanonical bundles and their direct images, J. Algebraic Geom. 27 (2018), no. 2, 211–272. MR 3764276, DOI https://doi.org/10.1090/jag/702
- Hossein Raufi, Singular hermitian metrics on holomorphic vector bundles, Ark. Mat. 53 (2015), no. 2, 359–382. MR 3391176
- Miles Reid, Nonnormal del Pezzo surfaces, Publ. Res. Inst. Math. Sci. 30 (1994), no. 5, 695–727. MR 1311389
- Eckart Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30, Springer-Verlag, Berlin, 1995. MR 1368632
- Qi Zhang, On projective varieties with nef anticanonical divisors, Math. Ann. 332 (2005), no. 3, 697–703. MR 2181470
Additional Information
Junyan Cao
Affiliation:
Université Paris 6, Institut de Mathématiques de Jussieu, 4, Place Jussieu, Paris 75252, France
MR Author ID:
1085653
Email:
junyan.cao@imj-prg.fr
Andreas Höring
Affiliation:
Université Côte d’Azur, CNRS, LJAD, 06108 Nice Cedex 2, France
Email:
Andreas.Hoering@unice.fr
Received by editor(s):
July 17, 2017
Received by editor(s) in revised form:
October 2, 2017
Published electronically:
April 25, 2019
Additional Notes:
This work was partially supported by the Agence Nationale de la Recherche grant project Foliage (ANR-16-CE40-0008) and the Agence Nationale de la Recherche grant “Convergence de Gromov-Hausdorff en géométrie kählérienne” (ANR-GRACK)
Dedicated:
Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday
Article copyright:
© Copyright 2019
University Press, Inc.
