Cycle conjectures and birational invariants over finite fields

• Samet Balkan ^[1] ; Stefan Schreieder ^[1]
  1. [1] Institute of Algebraic Geometry, Hannover, Germany
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 32, Nº. 2, 2026
• Idioma: inglés
• DOI: 10.1007/s00029-026-01142-0
• Enlaces
  • Texto completo
• Resumen

  • We study a natural birational invariant for varieties over finite fields and show that its vanishing on projective space is equivalent to the Tate conjecture, the Beilinson conjecture, and the Grothendieck–Serre semi-simplicity conjecture for all smooth projective varieties over finite fields. We further show that the Tate, Beilinson, and 1-semi-simplicity conjecture in half of the degrees implies those conjectures in all degrees.

• Referencias bibliográficas
  • Jannsen, U.: Mixed Motives and Algebraic K-Theory, LMN 1400. Springer, Berlin (1990)
  • Tate, J.: Endomorphisms of abelian varieties over finite fields. Invent. Math. 2, 134–144 (1966)
  • Artin, M., Swinnerton-Dyer, H.P.F.: The Shafarevich-Tate conjecture for pencils of elliptic curves on K3 surfaces. Invent. Math. 20, 249–266...
  • Rudakov, A.N., Shafarevich, I.R., Zink, T.: The influence of height on degenerations of algebraic surfaces of type K3. Math. USSR Izv. 20,...
  • Nygaard, N., Ogus, A.: Tate’s conjecture for K3 surfaces of finite height. Ann. of Math. 122, 461–507 (1985)
  • Maulik, D.: Supersingular K3 surfaces for large primes. Duke Math. J. 163, 2357–2425 (2014)
  • Charles, F.: The Tate conjecture for K3 surfaces over finite fields. Invent. Math. 194, 119–145 (2013)
  • Pera, K.M.: The Tate conjecture for K3 surfaces in odd characteristic. Invent. Math. 201, 625–668 (2015) Article Google Scholar
  • Kim, W., Pera, K.M.: 2-adic integral canonical models. Forum of Math., Sigma. 4, e28 (2016)
  • Tate, J.: Conjectures on algebraic cycles in l-adic cohomology, Motives. Proc. Sympos. Pure Math. 55(Part 1), 71–83 (1994). (Amer. Math. Soc.)
  • Totaro, B.: Recent progress on the Tate conjecture. Bulletin of the AMS 54, 575–590 (2017) Article Google Scholar
  • Geisser, T.: Tate’s conjecture, algebraic cycles and rational K-theory in characteristic p. K-Theory 13, 109–122 (1998)
  • Kahn, B.: A sheaf-theoretic reformulation of the Tate conjecture. Preprint arXiv:9801017 (1998)
  • Morrow, M.: A variational tate conjecture in crystalline cohomology. J. Eur. Math. Soc. 21, 3467–3511 (2019)
  • Soulé, C.: Groupes de Chow et K-théorie de variétés sur un corps fini. Math. Ann. 268, 317–345 (1984)
  • Kahn, B.: Équivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini. Ann. Sci. École Norm. Sup. 36, 977–1002...
  • Bloch, S., Ogus, A.: Gersten’s conjecture and the homology of schemes. Ann. Sci. Éc. Norm. Supér. 7, 181–201 (1974)
  • Colliot-Thélène, J.-L., Ojanguren, M.: Variétés unirationnelles non rationnelles: au-delà de l’exemple d’Artin et Mumford. Invent. Math. 97,...
  • Hassett, B., Pirutka, A., Tschinkel, Yu.: Stable rationality of quadric surface bundles over surfaces. Acta Math. 220, 341–365 (2018)
  • Schreieder, S.: Stably irrational hypersurfaces of small slopes. J. Amer. Math. Soc. 32, 1171–1199 (2019)
  • Colliot-Thélène, J.-L.: Birational invariants, purity and the Gersten conjecture, K-theory and algebraic geometry: connections with quadratic...
  • Schreieder, S.: Unramified cohomology, algebraic cycles and rationality. In: Farkas, G., et al. (eds.)
  • Rationality of Varieties, Progress in Mathematics, vol. 342, pp. 1–44. Birkhäuser (2021)
  • Cisinski, D.-C., Déglise, F.: Triangulated Categories of Mixed Motives. Springer (2019)
  • Moonen, B.: A remark on the Tate conjecture. J. Algebraic Geom. 28, 599–603 (2019)
  • Milne, J.: Values of zeta functions of varieties over finite fields. Amer. J. Math. 108, 297–360 (1986)
  • Beilinson, A.: Notes on absolute Hodge cohomology. Contemporary Math. AMS 55(Part I), 35–68 (1986)
  • de Jong, A.J.: Smoothness, semi-stability and alterations. Publ. Math. de l’IHÉS 83, 51–93 (1996)
  • Bhatt, B., Scholze, P.: The pro-étale topology of schemes. Astérisque 369, 99–201 (2015)
  • Jannsen, U.: Continuous Étale cohomology. Math. Ann. 280, 207–245 (1988)
  • Milne, J.S.: Étale Cohomology. Princeton University Press, Princeton, NJ (1980)
  • Schreieder, S.: Refined unramified cohomology of schemes. Compos. Math. 159, 1466–1530 (2023)
  • Schreieder, S.: A moving lemma for cohomology with support. arXiv:2207.08297, Preprint (2023)
  • Deligne, P.: Cohomologie étale, Lecture Notes in Mathematics, vol. 569. Springer-Verlag, Berlin (1977)
  • Deligne, P.: La conjecture de Weil, I. Publ. Math. I.H.É.S. 43, 273–307 (1974)
  • Deligne, P.: La conjecture de Weil, II. Publ. Math. I.H.É.S. 52, 137–252 (1980)
  • Colliot-Thélène, J.-L., Sansuc, J.-J., Soulé, C.: Torsion dans le groupe de Chow de codimension deux. Duke Math. J. 50, 763–801 (1983)
  • Bloch, S.: Algebraic cycles and higher K-theory. Adv. in Math. 61, 267–304 (1986)
  • Geisser, T., Levine, M.: The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky. J. Reine Angew. Math. 530, 55–103 (2001)
  • Beilinson, A.: Higher regulators and values of L-functions. J. Math. Sci. 30, 2036–2070 (1985)
  • Mazza, C., Voevodsky, V., Weibel, C.: Lecture Notes on Motivic Cohomology, vol. 2. American Mathematical Society, Clay Mathematics Monographs,...
  • Hartshorne, R.: Algebraic Geometry. Springer (1977)
  • Balkan, S.: Cohomological criteria for the Tate and Beilinson conjectures, PhD Thesis, Leibniz University Hannover (2024)