Endomorphisms of varieties and Bott vanishing
Authors:
Tatsuro Kawakami and Burt Totaro
Journal:
J. Algebraic Geom.
DOI:
https://doi.org/10.1090/jag/838
Published electronically:
November 6, 2024
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We show that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. This is a new way to analyze which varieties have nontrivial endomorphisms. In particular, we extend some classification results on varieties admitting endomorphisms (for Fano threefolds of Picard number one and several other cases) to any characteristic. The classification results in characteristic zero are due to Amerik–Rovinsky–Van de Ven, Hwang–Mok, Paranjape–Srinivas, Beauville, and Shao–Zhong. Our method also bounds the degree of morphisms into a given variety. Finally, we relate endomorphisms to global $F$-regularity.
References
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- Bruno Kahn, Sur le groupe des classes d’un schéma arithmétique, Bull. Soc. Math. France 134 (2006), no. 3, 395–415 (French, with English and French summaries). With an appendix by Marc Hindry. MR 2245999, DOI 10.24033/bsmf.2515
- Tatsuro Kawakami, On Kawamata-Viehweg type vanishing for three dimensional Mori fiber spaces in positive characteristic, Trans. Amer. Math. Soc. 374 (2021), no. 8, 5697–5717. MR 4293785, DOI 10.1090/tran/8369
- Timo Keller, On the $p$-torsion of the Tate-Shafarevich group of abelian varieties over higher dimensional bases over finite fields, J. Théor. Nombres Bordeaux 34 (2022), no. 2, 497–513 (English, with English and French summaries). MR 4524752
- Steven L. Kleiman, The Picard scheme, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 235–321. MR 2223410
- Kunihiko Kodaira, Complex manifolds and deformation of complex structures, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 283, Springer-Verlag, New York, 1986. Translated from the Japanese by Kazuo Akao; With an appendix by Daisuke Fujiwara. MR 815922, DOI 10.1007/978-1-4613-8590-5
- János Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács. MR 3057950, DOI 10.1017/CBO9781139547895
- Kazuhiro Konno, Generic Torelli theorem for hypersurfaces of certain compact homogeneous Kähler manifolds, Duke Math. J. 59 (1989), no. 1, 83–160. MR 1016881, DOI 10.1215/S0012-7094-89-05903-6
- A. G. Kuznetsov and Yu. G. Prokhorov, On higher-dimensional del Pezzo varieties, Izv. Ross. Akad. Nauk Ser. Mat. 87 (2023), no. 3, 75–148; English transl., Izv. Math. 87 (2023), no. 3, 488–561. MR 4640916, DOI 10.4213/im9385
- Ernst Kunz, Kähler differentials, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1986. MR 864975, DOI 10.1007/978-3-663-14074-0
- Robert Lazarsfeld, Some applications of the theory of positive vector bundles, Complete intersections (Acireale, 1983) Lecture Notes in Math., vol. 1092, Springer, Berlin, 1984, pp. 29–61. MR 775876, DOI 10.1007/BFb0099356
- 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, DOI 10.1007/978-3-642-18808-4
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- Sheng Meng, Building blocks of amplified endomorphisms of normal projective varieties, Math. Z. 294 (2020), no. 3-4, 1727–1747. MR 4074056, DOI 10.1007/s00209-019-02316-7
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, NJ, 1980. MR 559531
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- Shigeru Mukai, Biregular classification of Fano $3$-folds and Fano manifolds of coindex $3$, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 9, 3000–3002. MR 995400, DOI 10.1073/pnas.86.9.3000
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- Sheng Meng, De-Qi Zhang, and Guolei Zhong, Non-isomorphic endomorphisms of Fano threefolds, Math. Ann. 383 (2022), no. 3-4, 1567–1596. MR 4458408, DOI 10.1007/s00208-021-02274-8
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- K. H. Paranjape and V. Srinivas, Self-maps of homogeneous spaces, Invent. Math. 98 (1989), no. 2, 425–444. MR 1016272, DOI 10.1007/BF01388861
- N. I. Shepherd-Barron, Fano threefolds in positive characteristic, Compositio Math. 105 (1997), no. 3, 237–265. MR 1440723, DOI 10.1023/A:1000158618674
- Dennis M. Snow, Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces, Math. Ann. 276 (1986), no. 1, 159–176. MR 863714, DOI 10.1007/BF01450932
- The Stacks Project Authors, The Stacks Project, https://stacks.math.columbia.edu/, 2023.
- Karl Schwede and Karen E. Smith, Globally $F$-regular and log Fano varieties, Adv. Math. 224 (2010), no. 3, 863–894. MR 2628797, DOI 10.1016/j.aim.2009.12.020
- Feng Shao and Guolei Zhong, Boundedness of finite morphisms onto Fano manifolds with large Fano index, J. Algebra 639 (2024), 678–707. MR 4669541, DOI 10.1016/j.jalgebra.2023.10.030
- Hiromu Tanaka, Kawamata-Viehweg vanishing for toric varieties, Preprint, arXiv:2208.09680, 2022.
- Hiromu Tanaka, Fano threefolds in positive characteristic II, Preprint, arXiv:2308.08122, 2023.
- Hiromu Tanaka, Fano threefolds in positive characteristic IV, Preprint, arXiv:2308.08127, 2023.
- Burt Totaro, Endomorphisms of Fano 3-folds and log Bott vanishing, Preprint, arXiv:2305.18660, 2023, To appear in Math. Res. Lett.
- Burt Totaro, Bott vanishing for Fano threefolds, Math. Z. 307 (2024), no. 1, Paper No. 14, 31. MR 4735363, DOI 10.1007/s00209-024-03468-x
- P. M. H. Wilson, Fano fourfolds of index greater than one, J. Reine Angew. Math. 379 (1987), 172–181. MR 903639, DOI 10.1515/crll.1987.379.172
References
- Ekaterina Amerik, Maps onto certain Fano threefolds, Doc. Math. 2 (1997), 195–211. MR 1467127
- E. Yu. Amerik, Mappings onto quadrics, Mat. Zametki 81 (2007), no. 4, 621–624 (Russian); English transl., Math. Notes 81 (2007), no. 3-4, 549–552. MR 2352027, DOI 10.1134/S0001434607030327
- M. Artin, Supersingular $K3$ surfaces, Ann. Sci. École Norm. Sup. (4) 7 (1974), 543–567 (1975). MR 371899
- E. Amerik, M. Rovinsky, and A. Van de Ven, A boundedness theorem for morphisms between threefolds, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 2, 405–415 (English, with English and French summaries). MR 1697369, DOI 10.5802/aif.1679
- Piotr Achinger, Jakub Witaszek, and Maciej Zdanowicz, Global Frobenius liftability I, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 8, 2601–2648. MR 4269423, DOI 10.4171/jems/1063
- Piotr Achinger, Jakub Witaszek, and Maciej Zdanowicz, Global Frobenius liftability II: surfaces and Fano threefolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 24 (2023), no. 1, 329–366. MR 4587749, DOI 10.2422/2036-2145.202005_003
- Arnaud Beauville, Endomorphisms of hypersurfaces and other manifolds, Internat. Math. Res. Notices 1 (2001), 53–58. MR 1809497, DOI 10.1155/S1073792801000034
- Pierre Berthelot, Alexander Grothendieck, and Luc Illusie, Théorie des intersections et théorème de Riemann-Roch (SGA 6), Lecture Notes in Mathematics, vol. 225, Springer-Verlag, Berlin-New York, 1971.
- Bhargav Bhatt, Prismatic F-gauges, Princeton lecture notes, https://www.math.ias.edu/~bhatt/teaching/mat549f22/lectures.pdf, 2022.
- Michel Brion and Shrawan Kumar, Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, vol. 231, Birkhäuser Boston, Inc., Boston, MA, 2005. MR 2107324
- Anders Buch, Jesper F. Thomsen, Niels Lauritzen, and Vikram Mehta, The Frobenius morphism on a toric variety, Tohoku Math. J. (2) 49 (1997), no. 3, 355–366. MR 1464183, DOI 10.2748/tmj/1178225109
- Paolo Cascini, Sheng Meng, and De-Qi Zhang, Polarized endomorphisms of normal projective threefolds in arbitrary characteristic, Math. Ann. 378 (2020), no. 1-2, 637–665. MR 4150931, DOI 10.1007/s00208-019-01877-6
- A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020
- Torsten Ekedahl, Foliations and inseparable morphisms, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 139–149. MR 927978, DOI 10.1090/pspum/046.2/927978
- Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli, Fundamental algebraic geometry, Mathematical Surveys and Monographs, vol. 123, American Mathematical Society, Providence, RI, 2005. Grothendieck’s FGA explained. MR 2222646, DOI 10.1090/surv/123
- Andrea Fanelli and Stefan Schröer, The maximal unipotent finite quotient, unusual torsion in Fano threefolds, and exceptional Enriques surfaces, Épijournal Géom. Algébrique 4 (2020), Art. 11, 29 (English, with English and French summaries). MR 4149969, DOI 10.46298/epiga.2020.volume4.6151
- Takao Fujita, Vanishing theorems for semipositive line bundles, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 519–528. MR 726440, DOI 10.1007/BFb0099977
- Osamu Fujino, Multiplication maps and vanishing theorems for toric varieties, Math. Z. 257 (2007), no. 3, 631–641. MR 2328817, DOI 10.1007/s00209-007-0140-5
- Emmanuelle Garel, An extension of the trace map, J. Pure Appl. Algebra 32 (1984), no. 3, 301–313. MR 745360, DOI 10.1016/0022-4049(84)90094-X
- Frank Gounelas and Ariyan Javanpeykar, Invariants of Fano varieties in families, Mosc. Math. J. 18 (2018), no. 2, 305–319. MR 3831010, DOI 10.17323/1609-4514-2018-18-2-305-319
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 463157
- Jun-Muk Hwang and Ngaiming Mok, Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles, J. Algebraic Geom. 12 (2003), no. 4, 627–651. MR 1993759, DOI 10.1090/S1056-3911-03-00319-9
- V. A. Iskovskikh and Yu. G. Prokhorov, Fano varieties, Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1–247. MR 1668579
- Bruno Kahn, Sur le groupe des classes d’un schéma arithmétique, Bull. Soc. Math. France 134 (2006), no. 3, 395–415 (French, with English and French summaries). With an appendix by Marc Hindry. MR 2245999, DOI 10.24033/bsmf.2515
- Tatsuro Kawakami, On Kawamata-Viehweg type vanishing for three dimensional Mori fiber spaces in positive characteristic, Trans. Amer. Math. Soc. 374 (2021), no. 8, 5697–5717. MR 4293785, DOI 10.1090/tran/8369
- Timo Keller, On the $p$-torsion of the Tate-Shafarevich group of abelian varieties over higher dimensional bases over finite fields, J. Théor. Nombres Bordeaux 34 (2022), no. 2, 497–513 (English, with English and French summaries). MR 4524752
- Steven L. Kleiman, The Picard scheme, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 235–321. MR 2223410
- Kunihiko Kodaira, Complex manifolds and deformation of complex structures, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 283, Springer-Verlag, New York, 1986. Translated from the Japanese by Kazuo Akao; With an appendix by Daisuke Fujiwara. MR 815922, DOI 10.1007/978-1-4613-8590-5
- János Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács. MR 3057950, DOI 10.1017/CBO9781139547895
- Kazuhiro Konno, Generic Torelli theorem for hypersurfaces of certain compact homogeneous Kähler manifolds, Duke Math. J. 59 (1989), no. 1, 83–160. MR 1016881, DOI 10.1215/S0012-7094-89-05903-6
- A. G. Kuznetsov and Yu. G. Prokhorov, On higher-dimensional del Pezzo varieties, Izv. Ross. Akad. Nauk Ser. Mat. 87 (2023), no. 3, 75–148; English transl., Izv. Math. 87 (2023), no. 3, 488–561. MR 4640916, DOI 10.4213/im9385
- Ernst Kunz, Kähler differentials, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1986. MR 864975, DOI 10.1007/978-3-663-14074-0
- Robert Lazarsfeld, Some applications of the theory of positive vector bundles, Complete intersections (Acireale, 1983) Lecture Notes in Math., vol. 1092, Springer, Berlin, 1984, pp. 29–61. MR 775876, DOI 10.1007/BFb0099356
- 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, DOI 10.1007/978-3-642-18808-4
- G. Megyesi, Fano threefolds in positive characteristic, J. Algebraic Geom. 7 (1998), no. 2, 207–218. MR 1620094
- Sheng Meng, Building blocks of amplified endomorphisms of normal projective varieties, Math. Z. 294 (2020), no. 3-4, 1727–1747. MR 4074056, DOI 10.1007/s00209-019-02316-7
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, NJ, 1980. MR 559531
- Davesh Maulik and Bjorn Poonen, Néron-Severi groups under specialization, Duke Math. J. 161 (2012), no. 11, 2167–2206. MR 2957700, DOI 10.1215/00127094-1699490
- V. B. Mehta and V. Srinivas, Varieties in positive characteristic with trivial tangent bundle, Compositio Math. 64 (1987), no. 2, 191–212. With an appendix by Srinivas and M. V. Nori. MR 916481
- Shigeru Mukai, Biregular classification of Fano $3$-folds and Fano manifolds of coindex $3$, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 9, 3000–3002. MR 995400, DOI 10.1073/pnas.86.9.3000
- Sheng Meng and De-Qi Zhang, Normal projective varieties admitting polarized or int-amplified endomorphisms, Acta Math. Vietnam. 45 (2020), no. 1, 11–26. MR 4081361, DOI 10.1007/s40306-019-00333-6
- Sheng Meng, De-Qi Zhang, and Guolei Zhong, Non-isomorphic endomorphisms of Fano threefolds, Math. Ann. 383 (2022), no. 3-4, 1567–1596. MR 4458408, DOI 10.1007/s00208-021-02274-8
- Noboru Nakayama, Separable endomorphisms of surfaces in positive characteristic, Algebraic geometry in East Asia—Seoul 2008, Adv. Stud. Pure Math., vol. 60, Math. Soc. Japan, Tokyo, 2010, pp. 301–330. MR 2761933, DOI 10.2969/aspm/06010301
- Gianluca Occhetta and Jarosław A. Wiśniewski, On Euler-Jaczewski sequence and Remmert-van de Ven problem for toric varieties, Math. Z. 241 (2002), no. 1, 35–44. MR 1930984, DOI 10.1007/s002090100405
- K. H. Paranjape and V. Srinivas, Self-maps of homogeneous spaces, Invent. Math. 98 (1989), no. 2, 425–444. MR 1016272, DOI 10.1007/BF01388861
- N. I. Shepherd-Barron, Fano threefolds in positive characteristic, Compositio Math. 105 (1997), no. 3, 237–265. MR 1440723, DOI 10.1023/A:1000158618674
- Dennis M. Snow, Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces, Math. Ann. 276 (1986), no. 1, 159–176. MR 863714, DOI 10.1007/BF01450932
- The Stacks Project Authors, The Stacks Project, https://stacks.math.columbia.edu/, 2023.
- Karl Schwede and Karen E. Smith, Globally $F$-regular and log Fano varieties, Adv. Math. 224 (2010), no. 3, 863–894. MR 2628797, DOI 10.1016/j.aim.2009.12.020
- Feng Shao and Guolei Zhong, Boundedness of finite morphisms onto Fano manifolds with large Fano index, J. Algebra 639 (2024), 678–707. MR 4669541, DOI 10.1016/j.jalgebra.2023.10.030
- Hiromu Tanaka, Kawamata-Viehweg vanishing for toric varieties, Preprint, arXiv:2208.09680, 2022.
- Hiromu Tanaka, Fano threefolds in positive characteristic II, Preprint, arXiv:2308.08122, 2023.
- Hiromu Tanaka, Fano threefolds in positive characteristic IV, Preprint, arXiv:2308.08127, 2023.
- Burt Totaro, Endomorphisms of Fano 3-folds and log Bott vanishing, Preprint, arXiv:2305.18660, 2023, To appear in Math. Res. Lett.
- Burt Totaro, Bott vanishing for Fano threefolds, Math. Z. 307 (2024), no. 1, Paper No. 14, 31. MR 4735363, DOI 10.1007/s00209-024-03468-x
- P. M. H. Wilson, Fano fourfolds of index greater than one, J. Reine Angew. Math. 379 (1987), 172–181. MR 903639, DOI 10.1515/crll.1987.379.172
Additional Information
Tatsuro Kawakami
Affiliation:
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
MR Author ID:
1450194
ORCID:
0000-0002-8947-9665
Email:
tkawakami@math.kyoto-u.ac.jp
Burt Totaro
Affiliation:
Mathematics Department, UCLA, Los Angeles, California 90095-1555
MR Author ID:
272212
ORCID:
0000-0002-5573-4808
Email:
totaro@math.ucla.edu
Received by editor(s):
April 6, 2023
Received by editor(s) in revised form:
April 13, 2024, and July 2, 2024
Published electronically:
November 6, 2024
Additional Notes:
The first author was supported by JSPS KAKENHI Grant number JP22KJ1771 and JP24K16897. The second author was supported by NSF grant DMS-2054553, Simons Foundation grant SFI-MPS-SFM-00005512, and the Charles Simonyi Endowment at the Institute for Advanced Study.
Article copyright:
© Copyright 2024
University Press, Inc.
