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Incompressibility of Products of Pseudo-homogeneous Varieties

Published online by Cambridge University Press:  20 November 2018

Nikita A. Karpenko
Nikita A. Karpenko*
Affiliation:
Mathematical & Statistical Sciences, University of Alberta, Edmonton, AB e-mail: [email protected]
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Abstract

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We show that the conjectural criterion of $p$-incompressibility for products of projective homogeneous varieties in terms of the factors, previously known in a few special cases only, holds in general. Actually, the proof goes through for a wider class of varieties, including the norm varieties associated with symbols in Galois cohomology of arbitrary degree.

Keywords

20G1514C25algebraic groupsprojective homogeneous varietiesChow groups and motivescanonical dimension and incompressibility
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 4 , 01 December 2016 , pp. 824 - 833
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Berhuy, G. and Reichstein, Z., On the notion of canonical dimension for algebraic groups. Adv. Math. 198(2005), no. 1, 128–171. http://dx.doi.Org/10.1016/j.aim.2004.12.004 Google Scholar
[2] Biswas, I., Dhillon, A., and Hoffmann, N., On the essential dimension of coherent sheaves. arxiv:1306.6432Google Scholar
[3] Elman, R., Karpenko, N., and Merkurjev, A., The algebraic and geometric theory of quadratic forms. American Mathematical Society Colloquium Publications 56, American Mathematical Society, Providence, RI, 2008.Google Scholar
[4] Fulton, W., Intersection theory. Second ed. Ergebnisse der Mathematik und ihrer Grenzgebiete 2, Springer-Verlag, Berlin, 1998.Google Scholar
[5] Karpenko, N. A., Incompressibility of products by Grassmannians ofisotropic subspaces. http://www.ualberta.ca/∼karpenko/publ/pbg-r.pdf Google Scholar
[6] Karpenko, N. A., Canonical dimension. In: Proceedings of the International Congress of Mathematicians. II, New Delhi, Hindustan Book Agency, 2010. pp. 146–161.Google Scholar
[7] Karpenko, N. A., Sufficiently generic orthogonal Grassmannians. J. Algebra 372(2012), 365–375. http://dx.doi.Org/10.1016/j.jalgebra.2012.09.021 Google Scholar
[8] Karpenko, N. A., Upper motives of algebraic groups and incompressibility of Severi-Brauer varieties. J. Reine Angew. Math. 677(2013), 179–198.Google Scholar
[9] Karpenko, N. A., Incompressibility of products of Weil transfers of generalized Severi-Brauer varieties. Math. Z. (2014), 1–11. http://dx.doi.Org/10.1017/S1474748011000090 Google Scholar
[10] Karpenko, N. A. and Merkurjev, A. S., Canonical p-dimension of algebraic groups. Adv. Math. 205(2006), no. 2, 410–433. http://dx.doi.Org/10.1016/j.aim.2005.07.013 Google Scholar
[11] Karpenko, N. A. and Merkurjev, A. S., Essential dimension of finite p-groups. Invent. Math. 172(2008), no. 3, 491–508. http://dx.doi.Org/10.1007/s00222-007-0106-6 Google Scholar
[12] Karpenko, N. A. and Merkurjev, A. S., On standard norm varieties. Ann. Sci. Ec. Norm. Super. (4) 46(2013), no. 1,175-214.Google Scholar
[13] Karpenko, N. A. and Merkurjev, A. S., Motivic decomposition of compactifications of certain group varieties. J. Reine Angew. Math., to appear. http://dx.doi.Org/10.1515/crelle-2O16-0015 Google Scholar
[14] Karpenko, N. A. and Reichstein, Z., A numerical invariant for linear representations of finite groups. Comment. Math. Helv. 90(2015), no. 3, 667–701. With an appendix by Julia Pevtsova and Z. Reichstein. http://dx.doi.Org/10.41 71/CMH/367 Google Scholar
[15] Lotscher, R., MacDonald, M., Meyer, A., and Reichstein, Z., Essential dimension of algebraic tori. J. Reine Angew. Math. 677(2013), 1–13. http://dx.doi.Org/10.1515/crelle.2O12.010 Google Scholar
[16] Lotscher, R., MacDonald, M., Meyer, A., and Reichstein, Z., Essential p-dimension of algebraic groups whose connected component is a torus. Algebra Number Theory 7(2013), no. 8, 1817–1840. http://dx.doi.Org/10.2140/ant.2013.7.1817 Google Scholar
[17] Merkurjev, A. S., Essential dimension. In: Quadratic Forms: Algebra, Arithmetic, and Geometry. Contemp. Math. 493, American Mathematical Society, Providence, RI, 2009, pp. 299–326.Google Scholar
[18] Merkurjev, A. S., Essential dimension: a survey. Transform. Groups 18(2013), no. 2, 415–481. http://dx.doi.Org/10.1007/s00031-013-92 6-y Google Scholar
[19] Vishik, A. and Zainoulline, K., Motivic splitting lemma. Doc. Math. 13(2008), 81–96.Google Scholar