Chern classes of crystals

Esnault, Hélène; Shiho, Atsushi

doi:10.1090/tran/7342

Chern classes of crystals
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by Hélène Esnault and Atsushi Shiho PDF

Trans. Amer. Math. Soc. 371 (2019), 1333-1358 Request permission

Abstract:

The crystalline Chern classes of the value of a locally free crystal vanish on a smooth variety defined over a perfect field. Out of this we conclude new cases of de Jong’s conjecture relating the geometric étale fundamental group of a smooth projective variety defined over an algebraically closed field and the constancy of its category of isocrystals. We also discuss the case of the Gauß–Manin convergent isocrystal.

References

Tomoyuki Abe, Langlands correspondence for isocrystals and the existence of crystalline companions for curves, J. Amer. Math. Soc. 31 (2018), no. 4, 921–1057. MR 3836562, DOI 10.1090/jams/898
Fabrizio Andreatta and Luca Barbieri-Viale, Crystalline realizations of 1-motives, Math. Ann. 331 (2005), no. 1, 111–172. MR 2107442, DOI 10.1007/s00208-004-0576-4
Masanori Asakura and Kanetomo Sato, Chern class and Riemann-Roch theorem for cohomology theory without homotopy invariance, arXiv:1301.5829 (2013).
Pierre Berthelot, Cohomologie cristalline des schémas de caractéristique $p>0$, Lecture Notes in Mathematics, Vol. 407, Springer-Verlag, Berlin-New York, 1974 (French). MR 0384804
Pierre Berthelot, ${\scr D}$-modules arithmétiques. I. Opérateurs différentiels de niveau fini, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 2, 185–272 (French, with English summary). MR 1373933, DOI 10.24033/asens.1739
Pierre Berthelot, Dualité de Poincaré et formule de Künneth en cohomologie rigide, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 5, 493–498 (French, with English and French summaries). MR 1692313, DOI 10.1016/S0764-4442(97)88895-7
Pierre Berthelot, Finitude et pureté cohomologique en cohomologie rigide, Invent. Math. 128 (1997), no. 2, 329–377 (French). With an appendix in English by Aise Johan de Jong. MR 1440308, DOI 10.1007/s002220050143
Pierre Berthelot and Luc Illusie, Classes de Chern en cohomologie cristalline, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1695–A1697; ibid. A1750–A1752 (French). MR 269660
Pierre Berthelot and Arthur Ogus, Notes on crystalline cohomology, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. MR 0491705
Shiing-shen Chern, Differential geometry of fiber bundles, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, Amer. Math. Soc., Providence, R.I., 1952, pp. 397–411. MR 0045432
Bruno Chiarellotto and Bernard Le Stum, Sur la pureté de la cohomologie cristalline, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 8, 961–963 (French, with English and French summaries). MR 1649945, DOI 10.1016/S0764-4442(98)80122-5
Bruno Chiarellotto and Nobuo Tsuzuki, Cohomological descent of rigid cohomology for étale coverings, Rend. Sem. Mat. Univ. Padova 109 (2003), 63–215. MR 1997987
Richard M. Crew, Etale $p$-covers in characteristic $p$, Compositio Math. 52 (1984), no. 1, 31–45. MR 742696
Richard Crew, $F$-isocrystals and $p$-adic representations, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 111–138. MR 927977, DOI 10.1090/pspum/046.2/927977
Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77 (French). MR 498552, DOI 10.1007/BF02685881
Hélène Esnault, Characteristic classes of flat bundles, Topology 27 (1988), no. 3, 323–352. MR 963635, DOI 10.1016/0040-9383(88)90014-6
Hélène Esnault, Characteristic classes of flat bundles. II, $K$-Theory 6 (1992), no. 1, 45–56. MR 1186773, DOI 10.1007/BF00961334
Hélène Esnault, Algebraic differential characters, Regulators in analysis, geometry and number theory, Progr. Math., vol. 171, Birkhäuser Boston, Boston, MA, 2000, pp. 89–115. MR 1724888
Hélène Esnault and Atsushi Shiho, Convergent isocrystals on simply connected varieties, Ann. Inst. Fourier. (to appear)
Eric M. Friedlander, Étale homotopy of simplicial schemes, Annals of Mathematics Studies, No. 104, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 676809
Michel Gros, Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique, Mém. Soc. Math. France (N.S.) 21 (1985), 87 (French, with English summary). MR 844488
Michel Gros, Régulateurs syntomiques et valeurs de fonctions $L$ $p$-adiques. I, Invent. Math. 99 (1990), no. 2, 293–320 (French). With an appendix by Masato Kurihara. MR 1031903, DOI 10.1007/BF01234421
Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. MR 0354651
Luc Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 4, 501–661 (French). MR 565469, DOI 10.24033/asens.1374
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
Kiran S. Kedlaya, Full faithfulness for overconvergent $F$-isocrystals, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 819–835. MR 2099088
Adrian Langer, Semistable modules over Lie algebroids in positive characteristic, Doc. Math. 19 (2014), 509–540. MR 3218782
Adrian Langer, Bogomolov’s inequality for Higgs sheaves in positive characteristic, Invent. Math. 199 (2015), no. 3, 889–920. MR 3314517, DOI 10.1007/s00222-014-0534-z
Christopher Lazda, Incarnations of Berthelot’s conjecture, J. Number Theory 166 (2016), 137–157. MR 3486269, DOI 10.1016/j.jnt.2016.02.028
Lasse Peter Mannisto, Albanese and Picard 1-Motives in Positive Characteristic, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–University of California, Berkeley. MR 3295332
N. Markaryan, Crystalline characteristic classes via the Atiyah gerbe and generalized Chern–Weil construction, unpublished.
Arthur Ogus, $F$-isocrystals and de Rham cohomology. II. Convergent isocrystals, Duke Math. J. 51 (1984), no. 4, 765–850. MR 771383, DOI 10.1215/S0012-7094-84-05136-6
Arthur Ogus, $F$-crystals, Griffiths transversality, and the Hodge decomposition, Astérisque 221 (1994), ii+183. MR 1280543
A. Ogus and V. Vologodsky, Nonabelian Hodge theory in characteristic $p$, Publ. Math. Inst. Hautes Études Sci. 106 (2007), 1–138. MR 2373230, DOI 10.1007/s10240-007-0010-z
Bernard Saint-Donat, Techniques de descente cohomologique, Lecture Notes in Math., ch. Vbis, Springer Verlag, 1972.
I. Shafarevitch, On $p$-extensions, Rec. Math. [Mat. Sbornik] N.S. 20(62) (1947), 351–363 (Russian, with English summary). MR 0020546
Bernard Le Stum, Rigid cohomology, Cambridge Tracts in Mathematics, vol. 172, Cambridge University Press, Cambridge, 2007. MR 2358812, DOI 10.1017/CBO9780511543128
John Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144. MR 206004, DOI 10.1007/BF01404549