The Tate conjecture for even dimensional Gushel–Mukai varieties in characteristic $p\geq 5$
Authors:
Lie Fu and Ben Moonen
Journal:
J. Algebraic Geom.
DOI:
https://doi.org/10.1090/jag/836
Published electronically:
November 15, 2024
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We study Gushel–Mukai (GM) varieties of dimension $4$ or $6$ in characteristic $p$. Our main result is the Tate conjecture for all such varieties over finitely generated fields of characteristic $p\geq 5$. In the case of GM sixfolds, we follow the method used by Madapusi Pera in his proof of the Tate conjecture for K3 surfaces. As input for this, we prove a number of basic results about GM sixfolds, such as the fact that there are no nonzero global vector fields. For GM fourfolds, we prove the Tate conjecture by reducing it to the case of GM sixfolds by making use of the notion of generalized partners plus the fact that generalized partners in characteristic $0$ have isomorphic Chow motives in middle degree. Several steps in the proofs rely on results in characteristic $0$ that are proven in our paper [Épijournal Géom. Algébrique Volume spécial en l’honneur de Claire Voisin (2024), Paper No. 17].
References
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- Olivier Debarre, Atanas Iliev, and Laurent Manivel, On the period map for prime Fano threefolds of degree 10, J. Algebraic Geom. 21 (2012), no. 1, 21–59. MR 2846678, DOI 10.1090/S1056-3911-2011-00594-8
- O. Debarre, A. Iliev, and L. Manivel, Special prime Fano fourfolds of degree 10 and index 2, Recent advances in algebraic geometry, London Math. Soc. Lecture Note Ser., vol. 417, Cambridge Univ. Press, Cambridge, 2015, pp. 123–155. MR 3380447
- Olivier Debarre and Alexander Kuznetsov, Gushel-Mukai varieties: classification and birationalities, Algebr. Geom. 5 (2018), no. 1, 15–76. MR 3734109, DOI 10.14231/ag-2018-002
- Olivier Debarre and Alexander Kuznetsov, Gushel-Mukai varieties: linear spaces and periods, Kyoto J. Math. 59 (2019), no. 4, 897–953. MR 4032203, DOI 10.1215/21562261-2019-0030
- Olivier Debarre and Alexander Kuznetsov, Gushel-Mukai varieties: moduli, Internat. J. Math. 31 (2020), no. 2, 2050013, 59. MR 4083237, DOI 10.1142/S0129167X20500135
- Olivier Debarre and Alexander Kuznetsov, Gushel-Mukai varieties: intermediate Jacobians, Épijournal Géom. Algébrique 4 (2020), Art. 19, 45 (English, with English and French summaries). MR 4191422, DOI 10.46298/epiga.2020.volume4.6475
- Pierre Deligne, La conjecture de Weil pour les surfaces $K3$, Invent. Math. 15 (1972), 206–226 (French). MR 296076, DOI 10.1007/BF01404126
- P. Deligne, Relèvement des surfaces $K3$ en caractéristique nulle, Algebraic surfaces (Orsay, 1976–78) Lecture Notes in Math., vol. 868, Springer, Berlin, 1981, pp. 58–79 (French). Prepared for publication by Luc Illusie. MR 638598
- P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 79–297 (French). MR 1012168, DOI 10.1007/978-1-4613-9649-9_{3}
- Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982. MR 654325
- Pierre Deligne and Luc Illusie, Relèvements modulo $p^2$ et décomposition du complexe de de Rham, Invent. Math. 89 (1987), no. 2, 247–270 (French). MR 894379, DOI 10.1007/BF01389078
- M. Demazure, Automorphismes et déformations des variétés de Borel, Invent. Math. 39 (1977), no. 2, 179–186. MR 435092, DOI 10.1007/BF01390108
- Lie Fu and Ben Moonen, Algebraic cycles on Gushel–Mukai varieties, Épijournal de Géométrie Algébrique Volume spécial en l’honneur de Claire Voisin (2024), Paper No. 17., DOI 10.46298/epiga.2024.9815
- William Fulton, Intersection theory, 2nd ed., 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. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
- Atanas Iliev and Laurent Manivel, Fano manifolds of degree ten and EPW sextics, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 3, 393–426 (English, with English and French summaries). MR 2839455, DOI 10.24033/asens.2146
- Uwe Jannsen, Mixed motives and algebraic $K$-theory, Lecture Notes in Mathematics, vol. 1400, Springer-Verlag, Berlin, 1990. With appendices by S. Bloch and C. Schoen. MR 1043451, DOI 10.1007/BFb0085080
- Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
- Nicholas M. Katz and William Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77. MR 332791, DOI 10.1007/BF01405203
- Wansu Kim and Keerthi Madapusi Pera, 2-adic integral canonical models, Forum Math. Sigma 4 (2016), Paper No. e28, 34. MR 3569319, DOI 10.1017/fms.2016.23
- Mark Kisin, Integral models for Shimura varieties of abelian type, J. Amer. Math. Soc. 23 (2010), no. 4, 967–1012. MR 2669706, DOI 10.1090/S0894-0347-10-00667-3
- Mark Kisin, $\textrm {mod}\,p$ points on Shimura varieties of abelian type, J. Amer. Math. Soc. 30 (2017), no. 3, 819–914. MR 3630089, DOI 10.1090/jams/867
- Alexander Kuznetsov and Alexander Perry, Derived categories of Gushel-Mukai varieties, Compos. Math. 154 (2018), no. 7, 1362–1406. MR 3826460, DOI 10.1112/s0010437x18007091
- Keerthi Madapusi Pera, The Tate conjecture for K3 surfaces in odd characteristic, Invent. Math. 201 (2015), no. 2, 625–668. MR 3370622, DOI 10.1007/s00222-014-0557-5
- Keerthi Madapusi Pera, Integral canonical models for spin Shimura varieties, Compos. Math. 152 (2016), no. 4, 769–824. MR 3484114, DOI 10.1112/S0010437X1500740X
- Davesh Maulik, Supersingular K3 surfaces for large primes, Duke Math. J. 163 (2014), no. 13, 2357–2425. With an appendix by Andrew Snowden. MR 3265555, DOI 10.1215/00127094-2804783
- J. S. Milne, Shimura varieties and motives, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 447–523. MR 1265562, DOI 10.1090/pspum/055.2/1265562
- Ben Moonen, Models of Shimura varieties in mixed characteristics, Galois representations in arithmetic algebraic geometry (Durham, 1996) London Math. Soc. Lecture Note Ser., vol. 254, Cambridge Univ. Press, Cambridge, 1998, pp. 267–350. MR 1696489, DOI 10.1017/CBO9780511662010.008
- V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
- N. O. Nygaard, The Tate conjecture for ordinary $K3$ surfaces over finite fields, Invent. Math. 74 (1983), no. 2, 213–237. MR 723215, DOI 10.1007/BF01394314
- Niels Nygaard and Arthur Ogus, Tate’s conjecture for $K3$ surfaces of finite height, Ann. of Math. (2) 122 (1985), no. 3, 461–507. MR 819555, DOI 10.2307/1971327
- Kieran G. O’Grady, Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics, Duke Math. J. 134 (2006), no. 1, 99–137. MR 2239344, DOI 10.1215/S0012-7094-06-13413-0
- Kieran G. O’Grady, EPW-sextics: taxonomy, Manuscripta Math. 138 (2012), no. 1-2, 221–272. MR 2898755, DOI 10.1007/s00229-011-0472-7
- Alexander Perry, Laura Pertusi, and Xiaolei Zhao, Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties, Geom. Topol. 26 (2022), no. 7, 3055–3121. MR 4540901, DOI 10.2140/gt.2022.26.3055
- J. Piontkowski and A. Van de Ven, The automorphism group of linear sections of the Grassmannians $\textbf {G}(1,N)$, Doc. Math. 4 (1999), 623–664. MR 1719726
- Jordan Rizov, Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic, J. Reine Angew. Math. 648 (2010), 13–67. MR 2774304, DOI 10.1515/CRELLE.2010.078
- Ju. G. Zarhin, Abelian varieties in characteristic $p$, Mat. Zametki 19 (1976), no. 3, 393–400 (Russian). MR 422287
References
- Pierre Berthelot, Cohomologie cristalline des schémas de caractéristique $p>0$, Lecture Notes in Mathematics, Vol. 407, Springer-Verlag, Berlin-New York, 1974 (French). MR 384804
- Bhargav Bhatt, Matthew Morrow, and Peter Scholze, Integral $p$-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 128 (2018), 219–397. MR 3905467, DOI 10.1007/s10240-019-00102-z
- N. Bourbaki, Éléments de mathématique. Algèbre. Chapitres 1 à 3, Hermann, Paris, 1970 (French). MR 274237
- François Charles, The Tate conjecture for $K3$ surfaces over finite fields, Invent. Math. 194 (2013), no. 1, 119–145. MR 3103257, DOI 10.1007/s00222-012-0443-y
- François Charles, Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for $K3$ surfaces, and the Tate conjecture, Ann. of Math. (2) 184 (2016), no. 2, 487–526. MR 3548531, DOI 10.4007/annals.2016.184.2.4
- O. Debarre, Gushel–Mukai varieties, Preprint, arXiv:2001.03485v1, 2020.
- Olivier Debarre, Atanas Iliev, and Laurent Manivel, On the period map for prime Fano threefolds of degree 10, J. Algebraic Geom. 21 (2012), no. 1, 21–59. MR 2846678, DOI 10.1090/S1056-3911-2011-00594-8
- O. Debarre, A. Iliev, and L. Manivel, Special prime Fano fourfolds of degree 10 and index 2, Recent advances in algebraic geometry, London Math. Soc. Lecture Note Ser., vol. 417, Cambridge Univ. Press, Cambridge, 2015, pp. 123–155. MR 3380447
- Olivier Debarre and Alexander Kuznetsov, Gushel-Mukai varieties: classification and birationalities, Algebr. Geom. 5 (2018), no. 1, 15–76. MR 3734109, DOI 10.14231/ag-2018-002
- Olivier Debarre and Alexander Kuznetsov, Gushel-Mukai varieties: linear spaces and periods, Kyoto J. Math. 59 (2019), no. 4, 897–953. MR 4032203, DOI 10.1215/21562261-2019-0030
- Olivier Debarre and Alexander Kuznetsov, Gushel-Mukai varieties: moduli, Internat. J. Math. 31 (2020), no. 2, 2050013, 59. MR 4083237, DOI 10.1142/S0129167X20500135
- Olivier Debarre and Alexander Kuznetsov, Gushel-Mukai varieties: intermediate Jacobians, Épijournal Géom. Algébrique 4 (2020), Art. 19, 45 (English, with English and French summaries). MR 4191422, DOI 10.46298/epiga.2020.volume4.6475
- Pierre Deligne, La conjecture de Weil pour les surfaces $K3$, Invent. Math. 15 (1972), 206–226 (French). MR 296076, DOI 10.1007/BF01404126
- P. Deligne, Relèvement des surfaces $K3$ en caractéristique nulle, Algebraic surfaces (Orsay, 1976–78) Lecture Notes in Math., vol. 868, Springer, Berlin, 1981, pp. 58–79 (French). Prepared for publication by Luc Illusie. MR 638598
- P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois groups over ${\mathbf {Q}}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 79–297 (French). MR 1012168, DOI 10.1007/978-1-4613-9649-9_3
- Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982. MR 654325
- Pierre Deligne and Luc Illusie, Relèvements modulo $p^2$ et décomposition du complexe de de Rham, Invent. Math. 89 (1987), no. 2, 247–270 (French). MR 894379, DOI 10.1007/BF01389078
- M. Demazure, Automorphismes et déformations des variétés de Borel, Invent. Math. 39 (1977), no. 2, 179–186. MR 435092, DOI 10.1007/BF01390108
- Lie Fu and Ben Moonen, Algebraic cycles on Gushel–Mukai varieties, Épijournal de Géométrie Algébrique Volume spécial en l’honneur de Claire Voisin (2024), Paper No. 17., DOI 10.46298/epiga.2024.9815
- William Fulton, Intersection theory, 2nd ed., 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. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
- Atanas Iliev and Laurent Manivel, Fano manifolds of degree ten and EPW sextics, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 3, 393–426 (English, with English and French summaries). MR 2839455, DOI 10.24033/asens.2146
- Uwe Jannsen, Mixed motives and algebraic $K$-theory, Lecture Notes in Mathematics, vol. 1400, Springer-Verlag, Berlin, 1990. With appendices by S. Bloch and C. Schoen. MR 1043451, DOI 10.1007/BFb0085080
- Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
- Nicholas M. Katz and William Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77. MR 332791, DOI 10.1007/BF01405203
- Wansu Kim and Keerthi Madapusi Pera, 2-adic integral canonical models, Forum Math. Sigma 4 (2016), Paper No. e28, 34. MR 3569319, DOI 10.1017/fms.2016.23
- Mark Kisin, Integral models for Shimura varieties of abelian type, J. Amer. Math. Soc. 23 (2010), no. 4, 967–1012. MR 2669706, DOI 10.1090/S0894-0347-10-00667-3
- Mark Kisin, $\mod p$ points on Shimura varieties of abelian type, J. Amer. Math. Soc. 30 (2017), no. 3, 819–914. MR 3630089, DOI 10.1090/jams/867
- Alexander Kuznetsov and Alexander Perry, Derived categories of Gushel-Mukai varieties, Compos. Math. 154 (2018), no. 7, 1362–1406. MR 3826460, DOI 10.1112/s0010437x18007091
- Keerthi Madapusi Pera, The Tate conjecture for K3 surfaces in odd characteristic, Invent. Math. 201 (2015), no. 2, 625–668. MR 3370622, DOI 10.1007/s00222-014-0557-5
- Keerthi Madapusi Pera, Integral canonical models for spin Shimura varieties, Compos. Math. 152 (2016), no. 4, 769–824. MR 3484114, DOI 10.1112/S0010437X1500740X
- Davesh Maulik, Supersingular K3 surfaces for large primes, Duke Math. J. 163 (2014), no. 13, 2357–2425. With an appendix by Andrew Snowden. MR 3265555, DOI 10.1215/00127094-2804783
- J. S. Milne, Shimura varieties and motives, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 447–523. MR 1265562, DOI 10.1090/pspum/055.2/1265562
- Ben Moonen, Models of Shimura varieties in mixed characteristics, Galois representations in arithmetic algebraic geometry (Durham, 1996) London Math. Soc. Lecture Note Ser., vol. 254, Cambridge Univ. Press, Cambridge, 1998, pp. 267–350. MR 1696489, DOI 10.1017/CBO9780511662010.008
- V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
- N. O. Nygaard, The Tate conjecture for ordinary $K3$ surfaces over finite fields, Invent. Math. 74 (1983), no. 2, 213–237. MR 723215, DOI 10.1007/BF01394314
- Niels Nygaard and Arthur Ogus, Tate’s conjecture for $K3$ surfaces of finite height, Ann. of Math. (2) 122 (1985), no. 3, 461–507. MR 819555, DOI 10.2307/1971327
- Kieran G. O’Grady, Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics, Duke Math. J. 134 (2006), no. 1, 99–137. MR 2239344, DOI 10.1215/S0012-7094-06-13413-0
- Kieran G. O’Grady, EPW-sextics: taxonomy, Manuscripta Math. 138 (2012), no. 1-2, 221–272. MR 2898755, DOI 10.1007/s00229-011-0472-7
- Alexander Perry, Laura Pertusi, and Xiaolei Zhao, Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties, Geom. Topol. 26 (2022), no. 7, 3055–3121. MR 4540901, DOI 10.2140/gt.2022.26.3055
- J. Piontkowski and A. Van de Ven, The automorphism group of linear sections of the Grassmannians ${\mathbf {G}}(1,N)$, Doc. Math. 4 (1999), 623–664. MR 1719726
- Jordan Rizov, Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic, J. Reine Angew. Math. 648 (2010), 13–67. MR 2774304, DOI 10.1515/CRELLE.2010.078
- Ju. G. Zarhin, Abelian varieties in characteristic $p$, Mat. Zametki 19 (1976), no. 3, 393–400 (Russian). MR 422287
Additional Information
Lie Fu
Affiliation:
Département des mathématiques, IRMA & USIAS, Université de Strasbourg, 67000 Strasbourg, France
MR Author ID:
1016534
ORCID:
0000-0002-2177-3139
Email:
lie.fu@math.unistra.fr
Ben Moonen
Affiliation:
IMAPP, Radboud University Nijmegen, 6500GL Nijmegen, The Netherlands
MR Author ID:
254842
ORCID:
0000-0002-9467-5089
Email:
b.moonen@science.ru.nl
Received by editor(s):
November 21, 2022
Received by editor(s) in revised form:
March 26, 2024
Published electronically:
November 15, 2024
Additional Notes:
The first author was partially supported by the Radboud Excellence Initiative and by the Agence Nationale de la Recherche (ANR) under projects ANR-20-CE40-0023 and ANR-16-CE40-0011.
Article copyright:
© Copyright 2024
University Press, Inc.
