Weak positivity theorem and Frobenius stable canonical rings of geometric generic fibers
Author:
Sho Ejiri
Journal:
J. Algebraic Geom. 26 (2017), 691-734
DOI:
https://doi.org/10.1090/jag/698
Published electronically:
June 2, 2017
MathSciNet review:
3683424
Full-text PDF
Abstract |
References |
Additional Information
Abstract: In this paper, we prove the weak positivity theorem in positive characteristic when the canonical ring of the geometric generic fiber $F$ is finitely generated and the Frobenius stable canonical ring of $F$ is large enough. As its application, we show the subadditivity of Kodaira dimensions in some new cases.
References
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- Yujiro Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, J. Reine Angew. Math. 363 (1985), 1–46. MR 814013, DOI https://doi.org/10.1515/crll.1985.363.1
- Yujiro Kawamata, Semipositivity theorem for reducible algebraic fiber spaces, Pure Appl. Math. Q. 7 (2011), no. 4, Special Issue: In memory of Eckart Viehweg, 1427–1447. MR 2918168, DOI https://doi.org/10.4310/PAMQ.2011.v7.n4.a16
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- János Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235–268. MR 1064874
- J. Kollár and 14 coauthors, Flips and abundance for algebraic threefolds, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991. Astérisque 211 (1992), Société Mathématique de France, Paris, 1992.
- Herbert Lange and Ulrich Stuhler, Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe, Math. Z. 156 (1977), no. 1, 73–83 (German). MR 472827, DOI https://doi.org/10.1007/BF01215129
- 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
- Jun Lu, Mao Sheng, and Kang Zuo, An Arakelov inequality in characteristic $p$ and upper bound of $p$-rank zero locus, J. Number Theory 129 (2009), no. 12, 3029–3045. MR 2560851, DOI https://doi.org/10.1016/j.jnt.2009.05.015
- L. Moret-Bailly, Familles de courbes et de variétés abéliennes sur $\mathbb {P}^1$, Astérisque 86 (1981), 125–140.
- Lance Edward Miller and Karl Schwede, Semi-log canonical vs $F$-pure singularities, J. Algebra 349 (2012), 150–164. MR 2853631, DOI https://doi.org/10.1016/j.jalgebra.2011.08.035
- Tadao Oda, Vector bundles on an elliptic curve, Nagoya Math. J. 43 (1971), 41–72. MR 318151
- Zsolt Patakfalvi, Semi-positivity in positive characteristics, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 5, 991–1025 (English, with English and French summaries). MR 3294622, DOI https://doi.org/10.24033/asens.2232
- Z. Patakfalvi, On subadditivity of Kodaira dimension in positive characteristic, http://arxiv.org/abs/1308.5371 (2013).
- Z. Parakfalvi, K. Schwede, and W. Zhang, $F$-singularities in families, http://arxiv.org/pdf/1305.1646 (2013).
- Z. Patakfalvi, K. Schwede, and K. Tucker, Notes for the workshop on positive characteristic algebraic geometry, http://arxiv.org/abs/1412.2203v1 (2014).
- M. Raynaud, Contre-exemple au “vanishing theorem” en caractéristique $p>0$, C. P. Ramanujam—a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Springer, Berlin-New York, 1978, pp. 273–278 (French). MR 541027
- Karl Schwede, A canonical linear system associated to adjoint divisors in characteristic $p>0$, J. Reine Angew. Math. 696 (2014), 69–87. MR 3276163, DOI https://doi.org/10.1515/crelle-2012-0087
- Karl Schwede and Karen E. Smith, Globally $F$-regular and log Fano varieties, Adv. Math. 224 (2010), no. 3, 863–894. MR 2628797, DOI https://doi.org/10.1016/j.aim.2009.12.020
- Karen E. Smith, Globally F-regular varieties: applications to vanishing theorems for quotients of Fano varieties, Michigan Math. J. 48 (2000), 553–572. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786505, DOI https://doi.org/10.1307/mmj/1030132733
- L. Szpiro, Sur le théorème de rigidité de Parsin et Arakelov, Journées de Géométrie Algébrique de Rennes (Rennes, 1978) Astérisque, vol. 64, Soc. Math. France, Paris, 1979, pp. 169–202 (French). MR 563470
- L. Szpiro, Propriétés numériques du faisceau dualisant relatif, Astérisque 86 (1981), 44–78.
- Hiromu Tanaka, The X-method for klt surfaces in positive characteristic, J. Algebraic Geom. 24 (2015), no. 4, 605–628. MR 3383599, DOI https://doi.org/10.1090/S1056-3911-2014-00627-5
- Eckart Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 329–353. MR 715656, DOI https://doi.org/10.2969/aspm/00110329
- 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
- Qihong Xie, Counterexamples to the Kawamata-Viehweg vanishing on ruled surfaces in positive characteristic, J. Algebra 324 (2010), no. 12, 3494–3506. MR 2735396, DOI https://doi.org/10.1016/j.jalgebra.2010.09.009
References
- M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414–452. MR 0131423, DOI https://doi.org/10.1112/plms/s3-7.1.414
- M. Artin, Coverings of the rational double points in characteristic $p$, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 11–22. MR 0450263
- Lucian Bădescu, Algebraic surfaces, translated from the 1981 Romanian original by Vladimir Maşek and revised by the author, Universitext, Springer-Verlag, New York, 2001. MR 1805816
- E. Bombieri and D. Mumford, Enriques’ classification of surfaces in char. $p$. II, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 23–42. MR 0491719
- 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
- Yifei Chen and Lei Zhang, The subadditivity of the Kodaira dimension for fibrations of relative dimension one in positive characteristics, Math. Res. Lett. 22 (2015), no. 3, 675–696. MR 3350099, DOI https://doi.org/10.4310/MRL.2015.v22.n3.a3
- Brian Conrad, Grothendieck duality and base change, Lecture Notes in Mathematics, vol. 1750, Springer-Verlag, Berlin, 2000. MR 1804902
- O. Das and K. Schwede, The $F$-different and a canonical bundle formula, http://arxiv.org/abs/1508.07295 (2015).
- Shiro Goto and Keiichi Watanabe, The structure of one-dimensional $F$-pure rings, J. Algebra 49 (1977), no. 2, 415–421. MR 0453729, DOI https://doi.org/10.1016/0021-8693%2877%2990250-2
- Richard Fedder, $F$-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), no. 2, 461–480. MR 701505, DOI https://doi.org/10.2307/1999165
- O. Fujino, Semipositivity theorems for moduli problems, http://arxiv.org/abs/ 1210.5784 (2012).
- O. Fujino, Notes on the weak positivity theorems, to appear in Adv. Stud. Pure Math. (2013).
- Osamu Fujino, Direct images of relative pluricanonical bundles, Algebr. Geom. 3 (2016), no. 1, 50–62. MR 3455420, DOI https://doi.org/10.14231/AG-2016-003
- Osamu Fujino and Taro Fujisawa, Variations of mixed Hodge structure and semipositivity theorems, Publ. Res. Inst. Math. Sci. 50 (2014), no. 4, 589–661. MR 3273305, DOI https://doi.org/10.4171/PRIMS/145
- Osamu Fujino, Taro Fujisawa, and Morihiko Saito, Some remarks on the semipositivity theorems, Publ. Res. Inst. Math. Sci. 50 (2014), no. 1, 85–112. MR 3167580, DOI https://doi.org/10.4171/PRIMS/125
- Takao Fujita, On Kähler fiber spaces over curves, J. Math. Soc. Japan 30 (1978), no. 4, 779–794. MR 513085, DOI https://doi.org/10.2969/jmsj/03040779
- Phillip A. Griffiths, Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 125–180. MR 0282990
- Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. MR 0222093
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Robin Hartshorne, Generalized divisors on Gorenstein schemes, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), 1994, pp. 287–339. MR 1291023, DOI https://doi.org/10.1007/BF00960866
- Christopher D. Hacon and Sándor J. Kovács, Classification of higher dimensional algebraic varieties, Oberwolfach Seminars, vol. 41, Birkhäuser Verlag, Basel, 2010. MR 2675555
- Christopher D. Hacon and Zsolt Patakfalvi, Generic vanishing in characteristic $p>0$ and the characterization of ordinary abelian varieties, Amer. J. Math. 138 (2016), no. 4, 963–998. MR 3538148, DOI https://doi.org/10.1353/ajm.2016.0031
- Shigeru Iitaka, Algebraic geometry. An introduction to birational geometry of algebraic varieties, Graduate Texts in Mathematics, vol. 76, North-Holland Mathematical Library, 24, Springer-Verlag, New York-Berlin, 1982. MR 637060
- Junmyeong Jang, Generic p-rank for semi-stable fibrations, Thesis (Ph.D.)–Purdue University, 2008, ProQuest LLC, Ann Arbor, MI. MR 2712795
- Junmyeong Jang, Semi-stable fibrations of generic $p$-rank 0, Math. Z. 264 (2010), no. 2, 271–277. MR 2574975, DOI https://doi.org/10.1007/s00209-008-0462-y
- Yujiro Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), no. 2, 253–276. MR 622451
- Yujiro Kawamata, Kodaira dimension of algebraic fiber spaces over curves, Invent. Math. 66 (1982), no. 1, 57–71. MR 652646, DOI https://doi.org/10.1007/BF01404756
- Yujiro Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, J. Reine Angew. Math. 363 (1985), 1–46. MR 814013, DOI https://doi.org/10.1515/crll.1985.363.1
- Yujiro Kawamata, Semipositivity theorem for reducible algebraic fiber spaces, Pure Appl. Math. Q. 7 (2011), no. 4, Special Issue: In memory of Eckart Viehweg, 1427–1447. MR 2918168, DOI https://doi.org/10.4310/PAMQ.2011.v7.n4.a16
- János Kollár, Subadditivity of the Kodaira dimension: fibers of general type, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 361–398. MR 946244
- János Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235–268. MR 1064874
- J. Kollár and 14 coauthors, Flips and abundance for algebraic threefolds, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991. Astérisque 211 (1992), Société Mathématique de France, Paris, 1992.
- Herbert Lange and Ulrich Stuhler, Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe, Math. Z. 156 (1977), no. 1, 73–83 (German). MR 0472827, DOI https://doi.org/10.1007/BF01215129
- 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
- Jun Lu, Mao Sheng, and Kang Zuo, An Arakelov inequality in characteristic $p$ and upper bound of $p$-rank zero locus, J. Number Theory 129 (2009), no. 12, 3029–3045. MR 2560851, DOI https://doi.org/10.1016/j.jnt.2009.05.015
- L. Moret-Bailly, Familles de courbes et de variétés abéliennes sur $\mathbb {P}^1$, Astérisque 86 (1981), 125–140.
- Lance Edward Miller and Karl Schwede, Semi-log canonical vs $F$-pure singularities, J. Algebra 349 (2012), 150–164. MR 2853631, DOI https://doi.org/10.1016/j.jalgebra.2011.08.035
- Tadao Oda, Vector bundles on an elliptic curve, Nagoya Math. J. 43 (1971), 41–72. MR 0318151
- Zsolt Patakfalvi, Semi-positivity in positive characteristics, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 5, 991–1025 (English, with English and French summaries). MR 3294622
- Z. Patakfalvi, On subadditivity of Kodaira dimension in positive characteristic, http://arxiv.org/abs/1308.5371 (2013).
- Z. Parakfalvi, K. Schwede, and W. Zhang, $F$-singularities in families, http://arxiv.org/pdf/1305.1646 (2013).
- Z. Patakfalvi, K. Schwede, and K. Tucker, Notes for the workshop on positive characteristic algebraic geometry, http://arxiv.org/abs/1412.2203v1 (2014).
- M. Raynaud, Contre-exemple au “vanishing theorem” en caractéristique $p>0$, C. P. Ramanujam—a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Springer, Berlin-New York, 1978, pp. 273–278 (French). MR 541027
- Karl Schwede, A canonical linear system associated to adjoint divisors in characteristic $p>0$, J. Reine Angew. Math. 696 (2014), 69–87. MR 3276163, DOI https://doi.org/10.1515/crelle-2012-0087
- Karl Schwede and Karen E. Smith, Globally $F$-regular and log Fano varieties, Adv. Math. 224 (2010), no. 3, 863–894. MR 2628797, DOI https://doi.org/10.1016/j.aim.2009.12.020
- Karen E. Smith, Globally F-regular varieties: applications to vanishing theorems for quotients of Fano varieties, Dedicated to William Fulton on the occasion of his 60th birthday, Michigan Math. J. 48 (2000), 553–572. MR 1786505, DOI https://doi.org/10.1307/mmj/1030132733
- L. Szpiro, Sur le théorème de rigidité de Parsin et Arakelov, Journées de Géométrie Algébrique de Rennes (Rennes, 1978) Astérisque, vol. 64, Soc. Math. France, Paris, 1979, pp. 169–202 (French). MR 563470
- L. Szpiro, Propriétés numériques du faisceau dualisant relatif, Astérisque 86 (1981), 44–78.
- Hiromu Tanaka, The X-method for klt surfaces in positive characteristic, J. Algebraic Geom. 24 (2015), no. 4, 605–628. MR 3383599, DOI https://doi.org/10.1090/S1056-3911-2014-00627-5
- Eckart Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 329–353. MR 715656
- 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
- Qihong Xie, Counterexamples to the Kawamata-Viehweg vanishing on ruled surfaces in positive characteristic, J. Algebra 324 (2010), no. 12, 3494–3506. MR 2735396, DOI https://doi.org/10.1016/j.jalgebra.2010.09.009
Additional Information
Sho Ejiri
Affiliation:
Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
Email:
ejiri@ms.u-tokyo.ac.jp
Received by editor(s):
August 10, 2015
Received by editor(s) in revised form:
August 31, 2016
Published electronically:
June 2, 2017
Additional Notes:
The author wishes to express his gratitude to his supervisor, Professor Shunsuke Takagi, for suggesting the problems in this paper, for answering many questions, and for much helpful advice. The author is also grateful to Professors Yifei Chen, Yoshinori Gongyo and Zsolt Patakfalvi for valuable comments and discussions. He would like to thank Professor Akiyoshi Sannai and Doctors Takeru Fukuoka, Kenta Sato, and Fumiaki Suzuki for useful comments. He also wishes to thank the referee for the careful reading and valuable suggestions. The author was supported by JSPS KAKENHI grant No. 15J09117 and the Program for Leading Graduate Schools, MEXT, Japan
Article copyright:
© Copyright 2017
University Press, Inc.