Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T07:38:27.475Z Has data issue: false hasContentIssue false
  • English
  • Français

DISTINGUISHED MODELS OF INTERMEDIATE JACOBIANS

Part of: Abelian varieties and schemes Cycles and subschemes Arithmetic algebraic geometry

Published online by Cambridge University Press:  08 June 2018

Jeffrey D. Achter ,
Sebastian Casalaina-Martin  and
Charles Vial
Jeffrey D. Achter
Affiliation:
Colorado State University, Department of Mathematics, Fort Collins, CO 80523, USA (j.achter@colostate.edu)
Sebastian Casalaina-Martin
Affiliation:
University of Colorado, Department of Mathematics, Boulder, CO 80309, USA (casa@math.colorado.edu)
Charles Vial
Affiliation:
Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, D-33501, Germany (vial@math.uni-bielefeld.de)
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the image of the Abel–Jacobi map admits functorially a model over the field of definition, with the property that the Abel–Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also recover a result of Deligne on complete intersections of Hodge level 1.

Keywords

14-XX Algebraic Geometry

MSC classification

Primary: 14K30: Picard schemes, higher Jacobians 14C25: Algebraic cycles 14C30: Transcendental methods, Hodge theory , Hodge conjecture 11G10: Abelian varieties of dimension $> 1$ 11G35: Varieties over global fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 3 , May 2020 , pp. 891 - 918
Copyright
© Cambridge University Press 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The first author was partially supported by grants from the NSA (H98230-14-1-0161, H98230-15-1-0247 and H98230-16-1-0046). The second author was partially supported by a Simons Foundation Collaboration Grant for Mathematicians (317572) and NSA grant H98230-16-1-0053. The third author was supported by EPSRC Early Career Fellowship EP/K005545/1.

References

Achter, J. D., Casalaina-Martin, S. and Vial, C., On descending cohomology geometrically, Compos. Math. 153(7) (2017), 1446–1478. MR 3705264.Google Scholar
Achter, J. D., Casalaina-Martin, S. and Vial, C., Parameter spaces for algebraic equivalence, Int. Math. Res. Not. IMRN (2017), rnx178.Google Scholar
André, Y., Pour une théorie inconditionnelle des motifs, Publ. Math. Inst. Hautes Études Sci. (83) (1996), 5–49. MR 1423019.Google Scholar
Beauville, A., Quelques remarques sur la transformation de Fourier dans l’anneau de Chow d’une variété abélienne, in Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Mathematics, Volume 1016, pp. 238–260 (Springer, Berlin, 1983). MR 726428.Google Scholar
Bloch, S., Some elementary theorems about algebraic cycles on Abelian varieties, Invent. Math. 37(3) (1976), 215–228. MR 0429883.Google Scholar
Bloch, S., Torsion algebraic cycles and a theorem of Roitman, Compos. Math. 39(1) (1979), 107–127. MR 539002 (80k:14012).Google Scholar
Bloch, S. and Srinivas, V., Remarks on correspondences and algebraic cycles, Amer. J. Math. 105(5) (1983), 1235–1253. MR 714776 (85i:14002).Google Scholar
Charles, F. and Poonen, B., Bertini irreducibility theorems over finite fields, J. Amer. Math. Soc. 29(1) (2016), 81–94. MR 3402695.Google Scholar
Clemens, C. H. and Griffiths, P. A., The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356. MR 0302652.Google Scholar
Conrad, B., Chow’s K/k-image and K/k-trace, and the Lang–Néron theorem, Enseign. Math. (2) 52(1–2) (2006), 37–108. MR 2255529 (2007e:14068).Google Scholar
Deligne, P., Les intersections complètes de niveau de Hodge un, Invent. Math. 15 (1972), 237–250. MR 0301029 (46 #189).Google Scholar
Deligne, P., La conjecture de Weil. II, Publ. Math. Inst. Hautes Études Sci. (52) (1980), 137–252. MR 601520 (83c:14017).Google Scholar
Esnault, H. and Levine, M., Surjectivity of cycle maps, Astérisque (1993), (218), 203–226, Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). MR 1265315.Google Scholar
Griffiths, P. A., Periods of integrals on algebraic manifolds. II. Local study of the period mapping, Amer. J. Math. 90 (1968), 805–865. MR 0233825.Google Scholar
Griffiths, P. A., On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460–495. Ann. of Math. (2) 90 (1969), 496–541. MR 0260733.Google Scholar
Illusie, L., Miscellany on traces in ℓ-adic cohomology: a survey, Jpn. J. Math. 1(1) (2006), 107–136. MR 2261063 (2007g:14016).Google Scholar
Jannsen, U., Rigidity theorems for k- and h-cohomology and other functors, Preprint, 2015, arXiv:1503.08742 [math.AG].Google Scholar
Kleiman, S. L., Algebraic cycles and the Weil conjectures, in Dix exposés sur la cohomologie des schémas, pp. 359–386 (North-Holland, Amsterdam; Masson, Paris, 1968). MR 0292838 (45 #1920).Google Scholar
Kleiman, S. L., The Picard scheme, in Fundamental Algebraic Geometry, Mathematical Surveys and Monographs, Volume 123, pp. 235–321 (American Mathematical Society, Providence, RI, 2005). MR 2223410.Google Scholar
Lecomte, F., Rigidité des groupes de Chow, Duke Math. J. 53(2) (1986), 405–426. MR 850543 (88c:14013).Google Scholar
Mazur, B., Open problems: descending cohomology, geometrically, Not. Int. Congr. Chin. Math. 2(1) (2014), 37–40.Google Scholar
Milne, J. S., Abelian varieties (v2.00), 2008, available at http://www.jmilne.org/math/, p. 172.Google Scholar
Milne, J. S., Algebraic geometry (v6.02), 2017, available at http://www.jmilne.org/math/.Google Scholar
Murre, J. P., Applications of algebraic K-theory to the theory of algebraic cycles, in Algebraic Geometry, Sitges (Barcelona), 1983, Lecture Notes in Mathematics, Volume 1124, pp. 216–261 (Springer, Berlin, 1985). MR 805336 (87a:14006).Google Scholar
Otwinowska, A., Remarques sur les cycles de petite dimension de certaines intersections complètes, C. R. Acad. Sci. Paris Sér. I Math. 329(2) (1999), 141–146. MR 1710511.Google Scholar
Poonen, B., Bertini theorems over finite fields, Ann. of Math. (2) 160(3) (2004), 1099–1127. MR 2144974 (2006a:14035).Google Scholar
Rapoport, M., Complément à l’article de P. Deligne “La conjecture de Weil pour les surfaces K3”, Invent. Math. 15 (1972), 227–236. MR 0309943.Google Scholar
Rojtman, A. A., The torsion of the group of 0-cycles modulo rational equivalence, Ann. of Math. (2) 111(3) (1980), 553–569. MR 577137.Google Scholar
Weil, A., Sur les critères d’équivalence en géométrie algébrique, Math. Ann. 128 (1954), 95–127. MR 0065219 (16,398e).Google Scholar