On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces
Authors:
Sho Ejiri, Masataka Iwai and Shin-ichi Matsumura; with an appendix by Frédéric Touzet
Journal:
J. Algebraic Geom. 32 (2023), 477-517
DOI:
https://doi.org/10.1090/jag/814
Published electronically:
January 31, 2023
Full-text PDF
Abstract |
References |
Additional Information
Abstract: In this paper, we study the relative anti-canonical divisor $-K_{X/Y}$ of an algebraic fiber space $\phi \colon X\to Y$, and we reveal relations among positivity conditions of $-K_{X/Y}$, certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana–Cao–Matsumura’s equality on Hacon–McKernan’s question, whose original proof depends on analytics methods. The third result proves that algebraic fiber spaces with semi-ample relative anti-canonical divisor actually have a product structure via the base change by an appropriate finite étale cover of $Y$. Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.
References
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- Yujiro Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), no. 2, 253–276. MR 622451
- János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429–448. MR 1158625
- Robert Lazarsfeld, Positivity in algebraic geometry. I, 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. Classical setting: line bundles and linear series. MR 2095471, DOI 10.1007/978-3-642-18808-4
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- Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR 2104208
- Z. Patakfalvi and M. Zdanowicz, On the Beauville–Bogomolov decomposition in characteristic $p\ge 0$, arXiv:1912.12742, 2019.
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- 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 10.1090/jag/702
- Miles Reid, Nonnormal del Pezzo surfaces, Publ. Res. Inst. Math. Sci. 30 (1994), no. 5, 695–727. MR 1311389, DOI 10.2977/prims/1195165581
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References
- Florin Ambro, The moduli $b$-divisor of an lc-trivial fibration, Compos. Math. 141 (2005), no. 2, 385–403. MR 2134273, DOI 10.1112/S0010437X04001071
- 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, DOI 10.1215/00127094-2008-054
- B. Berndtsson and M. Păun, Bergman kernels and subadjunction, arXiv:1002.4145v1, 2010.
- Michel Brion, Some basic results on actions of nonaffine algebraic groups, Symmetry and spaces, Progr. Math., vol. 278, Birkhäuser Boston, Boston, MA, 2010, pp. 1–20. MR 2562620, DOI 10.1007/978-0-8176-4875-6_1
- 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
- Frédéric Campana, Junyan Cao, and S. Matsumura, Projective klt pairs with nef anti-canonical divisor, Algebr. Geom. 8 (2021), no. 4, 430–464. MR 4371536, DOI 10.14231/ag-2021-013
- Junyan Cao, Albanese maps of projective manifolds with nef anticanonical bundles, Ann. Sci. Éc. Norm. Supér. (4) 52 (2019), no. 5, 1137–1154 (English, with English and French summaries). MR 4057779, DOI 10.24033/asens.2405
- J. Cao, Positivité des images directes, extension d’Ohsawa-Takegoshi et applications, Habilitation thesis, Géométrie algébrique, Sorbonne University UPMC, 2019. fftel-02343266.
- JunYan Cao, Jean-Pierre Demailly, and S Matsumura, A general extension theorem for cohomology classes on non reduced analytic subspaces, Sci. China Math. 60 (2017), no. 6, 949–962. MR 3647124, DOI 10.1007/s11425-017-9066-0
- Junyan Cao and Andreas Höring, A decomposition theorem for projective manifolds with nef anticanonical bundle, J. Algebraic Geom. 28 (2019), no. 3, 567–597. MR 3959071, DOI 10.1090/jag/715
- Junyan Cao and Mihai Păun, Kodaira dimension of algebraic fiber spaces over abelian varieties, Invent. Math. 207 (2017), no. 1, 345–387. MR 3592759, DOI 10.1007/s00222-016-0672-6
- Ya Deng, Applications of the Ohsawa–Takegoshi extension theorem to direct image problems, Int. Math. Res. Not. IMRN 23 (2021), 17611–17633. MR 4349215, DOI 10.1093/imrn/rnaa018
- Yajnaseni Dutta and Takumi Murayama, Effective generation and twisted weak positivity of direct images, Algebra Number Theory 13 (2019), no. 2, 425–454. MR 3927051, DOI 10.2140/ant.2019.13.425
- Stéphane Druel, Some remarks on regular foliations with numerically trivial canonical class, Épijournal Géom. Algébrique 1 (2017), Art. 4, 20. MR 3743107, DOI 10.46298/epiga.2017.volume1.2057
- Lawrence Ein, Robert Lazarsfeld, Mircea Mustaţă, Michael Nakamaye, and Mihnea Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, 1701–1734 (English, with English and French summaries). MR 2282673
- Lawrence Ein, Robert Lazarsfeld, Mircea Mustaţă, Michael Nakamaye, and Mihnea Popa, Restricted volumes and base loci of linear series, Amer. J. Math. 131 (2009), no. 3, 607–651. MR 2530849, DOI 10.1353/ajm.0.0054
- Sho Ejiri and Yoshinori Gongyo, Nef anti-canonical divisors and rationally connected fibrations, Compos. Math. 155 (2019), no. 7, 1444–1456. MR 3975502, DOI 10.1112/s0010437x19007383
- Osamu Fujino, Fundamental theorems for semi log canonical pairs, Algebr. Geom. 1 (2014), no. 2, 194–228. MR 3238112, DOI 10.14231/AG-2014-011
- Osamu Fujino, Notes on the weak positivity theorems, Algebraic varieties and automorphism groups, Adv. Stud. Pure Math., vol. 75, Math. Soc. Japan, Tokyo, 2017, pp. 73–118. MR 3793363, DOI 10.2969/aspm/07510073
- Takao Fujita, On Kähler fiber spaces over curves, J. Math. Soc. Japan 30 (1978), no. 4, 779–794. MR 513085, DOI 10.2969/jmsj/03040779
- Osamu Fujino and Yoshinori Gongyo, On images of weak Fano manifolds II, Algebraic and complex geometry, Springer Proc. Math. Stat., vol. 71, Springer, Cham, 2014, pp. 201–207. MR 3278574, DOI 10.1007/978-3-319-05404-9_7
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- G. Hosono, M. Iwai, S. Matsumura, On projective manifolds with pseudo-effective tangent bundle, J. Inst. Math. Jussieu 21 (2022), no. 5, 1801–1830. MR 4476130
- Christopher D. Hacon and James McKernan, On Shokurov’s rational connectedness conjecture, Duke Math. J. 138 (2007), no. 1, 119–136. MR 2309156, DOI 10.1215/S0012-7094-07-13813-4
- 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] ©2018, pp. 143–195. MR 3832403, DOI 10.1090/conm/712/14346
- Andreas Höring, Positivity of direct image sheaves—a geometric point of view, Enseign. Math. (2) 56 (2010), no. 1-2, 87–142. MR 2674856, DOI 10.4171/LEM/56-1-4
- Yujiro Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), no. 2, 253–276. MR 622451
- János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429–448. MR 1158625
- Robert Lazarsfeld, Positivity in algebraic geometry. I, 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. Classical setting: line bundles and linear series. MR 2095471, DOI 10.1007/978-3-642-18808-4
- 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, DOI 10.1007/s00229-009-0322-z
- Frank Loray, Jorge Vitório Pereira, and Frédéric Touzet, Singular foliations with trivial canonical class, Invent. Math. 213 (2018), no. 3, 1327–1380. MR 3842065, DOI 10.1007/s00222-018-0806-0
- S. Matsumura and J. Wang, Structure theorem for projective klt pairs with nef anti-canonical divisor, arXiv:2105.14308v2, 2021.
- Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. MR 2104208
- Z. Patakfalvi and M. Zdanowicz, On the Beauville–Bogomolov decomposition in characteristic $p\ge 0$, arXiv:1912.12742, 2019.
- 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 10.24033/asens.2232
- 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 10.1090/jag/702
- Miles Reid, Nonnormal del Pezzo surfaces, Publ. Res. Inst. Math. Sci. 30 (1994), no. 5, 695–727. MR 1311389, DOI 10.2977/prims/1195165581
- 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 10.2969/aspm/00110329
- Juanyong Wang, On the Iitaka conjecture $C_{n,m}$ for Kähler fibre spaces, Ann. Fac. Sci. Toulouse Math. (6) 30 (2021), no. 4, 813–897 (English, with English and French summaries). MR 4350100, DOI 10.5802/afst.1690
Additional Information
Sho Ejiri
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka Metropolitan University, Osaka City, Osaka 558-8585, Japan
MR Author ID:
1225282
Email:
shoejiri.math@gmail.com
Masataka Iwai
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan
MR Author ID:
1359272
ORCID:
0000-0002-0273-0360
Email:
masataka.math@gmail.com, masataka@sci.osaka-cu.ac.jp
Shin-ichi Matsumura
Affiliation:
Mathematical Institute, Tohoku University, 6-3, Aramaki Aza-Aoba, Aoba-ku, Sendai 980-8578, Japan
MR Author ID:
1006398
ORCID:
0000-0002-5928-527X
Email:
mshinichi-math@tohoku.ac.jp, mshinichi0@gmail.com
Frédéric Touzet
ORCID:
0000-0002-0910-8451
Received by editor(s):
April 20, 2021
Received by editor(s) in revised form:
March 18, 2022
Published electronically:
January 31, 2023
Additional Notes:
The first author was supported by JSPS KAKENHI Grant $\sharp$18J00171. The second author was supported by JSPS KAKENHI Grant $\sharp$17J04457, by the Program for Leading Graduate Schools MEXT Japan, by the public interest incorporated foundation Fujukai, and by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849). The third author was supported by the Grant-in-Aid for Young Scientists (A) $\sharp$17H04821, Grant-in-Aid for Scientific Research (B) $\sharp$ 21H00976, and Fostering Joint International Research (A) $\sharp$19KK0342 from JSPS
Article copyright:
© Copyright 2023
University Press, Inc.
