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Nef anti-canonical divisors and rationally connected fibrations

Part of: Special varieties Families, fibrations

Published online by Cambridge University Press:  28 June 2019

Sho Ejiri  and
Yoshinori Gongyo
Sho Ejiri
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University Toyonaka, Osaka 560-0043, Japan email [email protected]
Yoshinori Gongyo
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan email [email protected]
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Abstract

We study the Iitaka–Kodaira dimension of nef relative anti-canonical divisors. As a consequence, we prove that given a complex projective variety with klt singularities, if the anti-canonical divisor is nef, then the dimension of a general fibre of the maximal rationally connected fibration is at least the Iitaka–Kodaira dimension of the anti-canonical divisor.

Keywords

anti-canonical divisorMRC fibration

MSC classification

Primary: 14D06: Fibrations, degenerations
Secondary: 14M22: Rationally connected varieties
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 7 , July 2019 , pp. 1444 - 1456
Copyright
© The Authors 2019 

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References

Abramovich, D. and Oort, F., Alterations and resolution of singularities , in Resolution of singularities (Springer, Basel, 2000), 39–108.10.1007/978-3-0348-8399-3_3Google Scholar
Bauer, T., Kovács, S. J., Küronya, A., Mistretta, E. C., Szemberg, T. and Urbinati, S., On positivity and base loci of vector bundles , Eur. J. Math. 1 (2015), 229–249.10.1007/s40879-015-0038-4Google Scholar
Boucksom, S., Demailly, J.-P., Pǎun, M. and Peternell, T., The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension , J. Algebraic Geom. 22 (2013), 201–248.10.1090/S1056-3911-2012-00574-8Google Scholar
Campana, F., Connexité rationnelle des variétés de Fano , Ann. Sci. Éc. Norm. Supér. (4) 25 (1992), 539–545.10.24033/asens.1658Google Scholar
Campana, F., Orbifolds, special varieties and classification theory , Ann. Inst. Fourier 54 (2004), 499–630.10.5802/aif.2027Google Scholar
Cao, J. and Höring, A., A decomposition theorem for projective manifolds with nef anticanonical bundle , J. Algebraic Geom. 28 (2019), 567–597.10.1090/jag/715Google Scholar
Chen, M. and Zhang, Q., On a question of Demailly–Peternell–Schneider , J. Eur. Math. Soc. 15 (2013), 1853–1858.10.4171/JEMS/406Google Scholar
de Jong, A. J., Smoothness, semi-stability and alterations , Publ. Math. Inst. Hautes Études Sci. 83 (1996), 51–93.Google Scholar
Druel, S., Quelques remarques sur la décomposition de Zariski divisorielle sur les variétés dont la première classe de Chern est nulle , Math. Z. 267 (2011), 413–423.10.1007/s00209-009-0626-4Google Scholar
Ejiri, S., Positivity of anti-canonical divisors and $F$ -purity of fibers, Preprint (2016),arXiv:1604.02022.Google Scholar
Ejiri, S., Weak positivity theorem and Frobenius stable canonical rings of geometric generic fibers , J. Algebraic Geom. 26 (2017), 691–734.10.1090/jag/698Google Scholar
Fujino, O., Notes on the weak positivity theorems , in Algebraic varieties and automorphism groups, Advanced Studies in Pure Mathematics, vol. 75 (Mathematical Society of Japan, Tokyo, 2017), 73–118.10.2969/aspm/07510073Google Scholar
Fujita, T., On Kähler fiber spaces over curves , J. Math. Soc. Japan 30 (1978), 779–794.10.2969/jmsj/03040779Google Scholar
Graber, T., Harris, J. and Starr, J., Families of rationally connected varieties , J. Amer. Math. Soc. 16 (2003), 57–67.10.1090/S0894-0347-02-00402-2Google Scholar
Hacon, C. D. and McKernan, J., Shokurov’s rational connectedness conjecture, Preprint (2005), arXiv:math/0504330.Google Scholar
Hacon, C. D. and McKernan, J., On Shokurov’s rational connectedness conjecture , Duke Math. J. 138 (2007), 119–136.10.1215/S0012-7094-07-13813-4Google Scholar
Hara, N. and Watanabe, K.-i., F-regular and F-pure rings vs. log terminal and log canonical singularities , J. Algebraic Geom. 11 (2002), 363–392.10.1090/S1056-3911-01-00306-XGoogle Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977).10.1007/978-1-4757-3849-0Google Scholar
Hochster, M. and Huneke, C., Tight closure and strong F-regularity , Mém. Soc. Math. Fr. 38 (1989), 119–133.Google Scholar
Hochster, M. and Roberts, J. L., The purity of the Frobenius and local cohomology , Adv. Math. 21 (1976), 117–172.10.1016/0001-8708(76)90073-6Google Scholar
Kawamata, Y., Characterization of abelian varieties , Compos. Math. 43 (1981), 253–276.Google Scholar
Kollár, J., Miyaoka, Y. and Mori, S., Rational connectedness and boundedness of Fano manifolds , J. Differential Geom. 36 (1992), 765–779.Google Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998).10.1017/CBO9780511662560Google Scholar
Nakayama, N., Zariski-decomposition and abundance, MSJ Memoirs, vol. 14 (Mathematical Society of Japan, Tokyo, 2004).10.2969/msjmemoirs/014010000Google Scholar
Patakfalvi, Z., Semi-positivity in positive characteristics , Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), 991–1025.10.24033/asens.2232Google Scholar
Patakfalvi, Z., Schwede, K. and Zhang, W., F-singularities in families , Algebr. Geom. 5 (2018), 264–327.Google Scholar
Schwede, K., Generalized test ideals, sharp F-purity, and sharp test elements , Math. Res. Lett. 15 (2008), 1251–1261.Google Scholar
Schwede, K. and Smith, K. E., Globally F-regular and log Fano varieties , Adv. Math. 224 (2010), 863–894.Google Scholar
Schwede, K. and Wenliang, Z., Bertini theorems for F-singularities , Proc. Lond. Math. Soc. (3) 107 (2013), 851–874.Google Scholar
Viehweg, E., Weak positivity and the additivity of the Kodaira dimension for certain fiber spaces , in Algebraic varieties and analytic varieties (Mathematical Society of Japan, Tokyo, 1983), 329–353.10.2969/aspm/00110329Google Scholar
Viehweg, E., Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 30 (Springer, Berlin, 1995).Google Scholar
Zhang, Q., On projective varieties with nef anticanonical divisors , Math. Ann. 332 (2005), 697–703.10.1007/s00208-005-0649-zGoogle Scholar
Zhang, Q., Rational connectedness of log Q-Fano varieties , J. Reine Angew. Math. 590 (2006), 131–142.Google Scholar