Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter November 28, 2015

Some remarks concerning Voevodsky’s nilpotence conjecture

Marcello Bernardara, Matilde Marcolli and Gonçalo Tabuada

Abstract

In this article we extend Voevodsky’s nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories. Making use of this noncommutative generalization, we then address Voevodsky’s original conjecture in the following cases: quadric fibrations, intersection of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, homological projective duals, and Moishezon manifolds.

Funding source: National Science Foundation

Award Identifier / Grant number: CAREER Award #1350472

Award Identifier / Grant number: DMS-1201512

Award Identifier / Grant number: PHY-1205440

Funding statement: G. Tabuada was partially supported by the National Science Foundation CAREER Award #1350472 and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project grant UID/MAT/00297/2013 (Centro de Matemática e Aplicações). M. Marcolli was partially supported by the NSF grants DMS-1201512 and PHY-1205440.

Acknowledgements

The authors are grateful to Bruno Kahn and Claire Voisin for useful comments and answers, as well as to the anonymous referee for useful remarks.

References

[1] Y. André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panor. Synthèses 17, Société Mathématique de France, Paris 2004. Search in Google Scholar

[2] Y. André and B. Kahn, Nilpotence, radicaux et structures monoïdales, Rend. Semin. Mat. Univ. Padova 108 (2002), 107–291. Search in Google Scholar

[3] M. Artin, Algebraization of formal moduli. II: Existence of modification, Ann. of Math. (2) 91 (1970), 88–135. 10.2307/1970602Search in Google Scholar

[4] M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. Lond. Math. Soc. (3) 25 (1972), 75–95. 10.1007/978-1-4757-4265-7_29Search in Google Scholar

[5] A. Auel, M. Bernardara and M. Bolognesi, Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems, J. Math. Pures Appl. (9) 102 (2014), 249–291. 10.1016/j.matpur.2013.11.009Search in Google Scholar

[6] H. Bass, Algebraic K-theory, W. A. Benjamin, New York 1968. Search in Google Scholar

[7] M. Bernardara, M. Bolognesi and D. Faenzi, Homological projective duality for determinantal varieties, preprint (2014), http://arxiv.org/abs/1410.7803. 10.1016/j.aim.2016.04.003Search in Google Scholar

[8] M. Bernardara and G. Tabuada, Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) noncommutative motives, preprint (2013), http://arxiv.org/abs/1310.6020. Search in Google Scholar

[9] A. Bondal and D. Orlov, Derived categories of coherent sheaves, International congress of mathematicians. Vol. II: Invited lectures (Beijing 2002), Higher Education Press, Beijing (2002), 47–56. Search in Google Scholar

[10] D.-C. Cisinski and G. Tabuada, Symmetric monoidal structure on noncommutative motives, J. K-Theory 9 (2012), no. 2, 201–268. 10.1017/is011011005jkt169Search in Google Scholar

[11] F. Cossec, Reye congruences, Trans. Amer. Math. Soc. 280 (1983), no. 2, 737–751. 10.1090/S0002-9947-1983-0716848-4Search in Google Scholar

[12] S. Gorchinskiy and V. Guletskii, Motives and representability of algebraic cycles on threefolds over a field, J. Algebraic Geom. 21 (2012), no. 2, 347–373. 10.1090/S1056-3911-2011-00548-1Search in Google Scholar

[13] T. Graber, J. Harris and J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), no. 1, 57–67. 10.1090/S0894-0347-02-00402-2Search in Google Scholar

[14] D. Huybrechts, Fourier–Mukai transforms in algebraic geometry, Oxford Math. Monogr., Clarendon Press, Oxford, 2006. 10.1093/acprof:oso/9780199296866.001.0001Search in Google Scholar

[15] C. Ingalls and A. Kuznetsov, On nodal Enriques surfaces and quartic double solids, preprint (2010), http://arxiv.org/abs/1012.3530. 10.1007/s00208-014-1066-ySearch in Google Scholar

[16] B. Kahn and R. Sebastian, Smash-nilpotent cycles on abelian 3-folds, Math. Res. Lett. 16 (2009), no. 6, 1007–1010. 10.4310/MRL.2009.v16.n6.a8Search in Google Scholar

[17] B. Keller, On differential graded categories, International congress of mathematicians. Vol. II: Invited lectures (Madrid 2006), European Mathematical Society, Zürich (2006), 151–190. 10.4171/022-2/8Search in Google Scholar

[18] M. Kontsevich, Noncommutative motives, Talk at the Institute for Advanced Study on the occasion of the 61st birthday of Pierre Deligne, Princeton, October 2005. Video available at http://video.ias.edu/Geometry-and-Arithmetic. Search in Google Scholar

[19] M. Kontsevich, Notes on motives in finite characteristic, Algebra, arithmetic, and geometry. In honor of Yu. I. Manin. Vol. II, Progr. Math. 270, Birkhäuser, Boston (2009), 213–247. 10.1007/978-0-8176-4747-6_7Search in Google Scholar

[20] M. Kontsevich, Mixed noncommutative motives, Talk at the workshop on homological mirror symmetry, Miami 2010. Notes available at https://math.berkeley.edu/~auroux/frg/miami10-notes/. Search in Google Scholar

[21] A. Kuznetsov, Homological projective duality for Grassmannians of lines, preprint (2006), http://arxiv.org/abs/math/0610957. Search in Google Scholar

[22] A. Kuznetsov, Homological projective duality, Publ. Math. Inst. Hautes Études Sci. 105 (2007), 157–220. 10.1007/s10240-007-0006-8Search in Google Scholar

[23] A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008), no. 5, 1340–1369. 10.1016/j.aim.2008.03.007Search in Google Scholar

[24] A. Kuznetsov, Semiorthogonal decompositions in algebraic geometry, preprint (2014), http://arxiv.org/abs/1404.3143. Search in Google Scholar

[25] V. Lunts and D. Orlov, Uniqueness of enhancement for triangulated categories, J. Amer. Math. Soc. 23 (2010), 853–908. 10.1090/S0894-0347-10-00664-8Search in Google Scholar

[26] M. Marcolli and G. Tabuada, Unconditional motivic Galois groups and Voevodsky’s nilpotence conjecture in the noncommutative world, preprint (2011), http://arxiv.org/abs/1112.5422v1. Search in Google Scholar

[27] M. Marcolli and G. Tabuada, Kontsevich’s noncommutative numerical motives, Compos. Math. 148 (2012), no. 6, 1811–1820. 10.1112/S0010437X12000383Search in Google Scholar

[28] M. Marcolli and G. Tabuada, Noncommutative motives and their applications, preprint (2013), http://arxiv.org/abs/1311.2867; to appear in MSRI publications, Berkeley. Search in Google Scholar

[29] M. Marcolli and G. Tabuada, Noncommutative motives, numerical equivalence, and semi-simplicity, Amer. J. Math. 136 (2014), no. 1, 59–75. 10.1353/ajm.2014.0004Search in Google Scholar

[30] T. Matsusaka, The criteria for algebraic equivalence and the torsion group, Amer. J. Math. 79 (1957), 53–66. 10.2307/2372383Search in Google Scholar

[31] B. G. Moishezon, On n-dimensional compact varieties with n algebraically independent meromorphic functions. I, II and III, Amer. Math. Soc. Transl. II 63 (1967), 51–177. 10.1090/trans2/063/02Search in Google Scholar

[32] G. Tabuada, A guided tour through the garden of noncommutative motives, Clay Math. Proc. 16 (2012), 259–276. Search in Google Scholar

[33] G. Tabuada, Chow motives versus noncommutative motives, J. Noncommut. Geom. 7 (2013), no. 3, 767–786. 10.4171/JNCG/134Search in Google Scholar

[34] G. Tabuada and M. Van den Bergh, Noncommutative motives of Azumaya algebras, J. Inst. Math. Jussieu 14 (2015), no. 2, 379–403. 10.1017/S147474801400005XSearch in Google Scholar

[35] C. Vial, Algebraic cycles and fibrations, Doc. Math. 18 (2013), 1521–1553. Search in Google Scholar

[36] V. Voevodsky, A nilpotence theorem for cycles algebraically equivalent to zero, Int. Math. Res. Not. IMRN 1995 (1995), no. 4, 187–198. 10.1155/S1073792895000158Search in Google Scholar

[37] C. Voisin, Remarks on zero-cycles of self-products of varieties, Moduli of vector bundles (Sanda 1994, Kyoto 1994), Lecture Notes in Pure and Appl. Math. 179, Dekker, New York (1996), 265–285. Search in Google Scholar

[38] S. Zube, Exceptional vector bundles on Enriques surfaces, Math. Notes 61 (1997), no. 6, 693–699. 10.1007/BF02361211Search in Google Scholar

Received: 2014-11-21
Published Online: 2015-11-28
Published in Print: 2018-5-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston