Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T08:40:29.639Z Has data issue: false hasContentIssue false
  • English
  • Français

Derived equivalent threefolds, algebraic representatives, and the coniveau filtration

Published online by Cambridge University Press:  08 May 2018

JEFFREY D. ACHTER ,
SEBASTIAN CASALAINA-MARTIN  and
CHARLES VIAL
JEFFREY D. ACHTER
Affiliation:
Colorado State University, Department of Mathematics, Fort Collins, CO 80523, U.S.A.
SEBASTIAN CASALAINA-MARTIN
Affiliation:
University of Colorado, Department of Mathematics, Boulder, CO 80309, U.S.A.
CHARLES VIAL
Affiliation:
Universität Bielefeld, Fakultät für Mathematik, Bielefeld 33501, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

A conjecture of Orlov predicts that derived equivalent smooth projective varieties over a field have isomorphic Chow motives. The conjecture is known for curves, and was recently observed for surfaces by Huybrechts. In this paper we focus on threefolds over perfect fields, and unconditionally secure results, which are implied by Orlov's conjecture, concerning the geometric coniveau filtration, and abelian varieties attached to smooth projective varieties.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 1 , July 2019 , pp. 123 - 131
Copyright
Copyright © Cambridge Philosophical Society 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.)

References

REFERENCES

[1] Achter, J. D., Casalaina–Martin, S., Honigs, K. and Vial, Ch. Appendix to: Derived equivalence, Albanese varieties, and the zeta functions of 3-dimensional varieties, by K. Honigs. Proc. Amer. Math. Soc. 146 (2018), no. 3, 1005–1013.Google Scholar
[2] Achter, J. D., Casalaina–Martin, S. and Vial, Ch. On descending cohomology geometrically. Compositio Math. 153 (2017), no. 7, 1446–1478.Google Scholar
[3] Achter, J. D., Casalaina–Martin, S. and Vial, Ch. Distinguished models of intermediate Jacobians. J. Inst. Math. Jussieu (to appear) ArXiv e-prints (2016), 1611.07471.Google Scholar
[4] Arapura, D. and Kang, S.-J. Functoriality of the coniveau filtration. Canad. Math. Bull. 50 (2007), no. 2, 161–171.Google Scholar
[5] Antieau, B., Krashen, D. and Ward, M. Derived categories of torsors for abelian schemes. Adv. Math. 306 (2017), 1–23.Google Scholar
[6] Bondal, A. and Orlov, D. Reconstruction of a variety from the derived category and groups of autoequivalences. Compositio Math. 125 (2001), no. 3, 327–344.Google Scholar
[7] Bernardara, M. and Tabuada, G. From semi-orthogonal decompositions to polarized intermediate Jacobians via Jacobians of noncommutative motives. Mosc. Math. J. 16 (2016), no. 2, 205–235.Google Scholar
[8] Curtis, C. W. and Reiner, I. Methods Of Representation Theory. Vol. I, Wiley Classics Library (John Wiley & Sons, Inc., New York, 1990) With applications to finite groups and orders, Reprint of the 1981 original, A Wiley-Interscience Publication.Google Scholar
[9] Déglise, F. Suite spectrale du coniveau et t-structure homotopique. Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), no. 3, 591–609.Google Scholar
[10] Honigs, K. Derived equivalent surfaces and abelian varieties, and their zeta functions. Proc. Amer. Math. Soc. 143 (2015), no. 10, 4161–4166.Google Scholar
[11] Honigs, K. Derived equivalence, Albanese varieties, and the zeta functions of 3-dimensional varieties. Proc. Amer. Math. Soc. 146 (2018), no. 3, 1005–1013.Google Scholar
[12] Huybrechts, D. Motives of derived equivalent K3 surfaces. Abh. Math. Semin. Univ. Hambg., to appear.Google Scholar
[13] Huybrechts, D. Fourier–Mukai transforms in algebraic geometry. Oxford Math. Monogr. (The Clarendon Press, Oxford University Press, Oxford, 2006).Google Scholar
[14] Lieblich, M. and Olsson, M. Fourier–Mukai partners of K3 surfaces in positive characteristic. Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 5, 1001–1033.Google Scholar
[15] Mazur, B. Open problems: Descending cohomology, geometrically. Not. Interna. Congress of Chinese Mathematicians 2 (2014), no. 1, 37–40.Google Scholar
[16] Murre, J. P. Applications of algebraic K-theory to the theory of algebraic cycles. Algebraic geometry (Sitges (Barcelona), 1983) Lecture Notes in Math. vol. 1124 (Springer, Berlin, 1985), pp. 216–261.Google Scholar
[17] Orlov, D. O. Derived categories of coherent sheaves and equivalences between them. Uspekhi Mat. Nauk 58 (2003), no. 3(351), 89–172.Google Scholar
[18] Orlov, D. O. Derived categories of coherent sheaves, and motives. Uspekhi Mat. Nauk 60 (2005), no. 6(366), 231–232.Google Scholar
[19] Popa, M. and Schnell, C. Derived invariance of the number of holomorphic 1-forms and vector fields. Ann. Sci. Ecole. Norm. Sup. (4) 44 (2011), no. 3, 527–536.Google Scholar
[20] Rouquier, R. Automorphismes, graduations et catégories triangulées. J. Inst. Math. Jussieu 10 (2011), no. 3, 713–751.Google Scholar