Comptes Rendus
Algebra/Algebraic Geometry
Incompressibility of orthogonal grassmannians
Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1131-1134.

We prove the following conjecture due to Bryant Mathews (2008). Let Q be the orthogonal grassmannian of totally isotropic i-planes of a non-degenerate quadratic form q over an arbitrary field (where i is an integer satisfying 1i(dimq)/2). If the degree of each closed point on Q is divisible by 2i and the Witt index of q over the function field of Q is equal to i, then the variety Q is 2-incompressible.

Nous démontrons la conjecture ci-dessous due à Bryant Mathews (2008). Soit Q la grassmannienne orthogonale des i-plans totalement isotropes dʼune forme quadratique non dégénérée q sur un corps arbitraire (où i est un entier satisfaisant 1i(dimq)/2). Si le degré de tout point fermé sur Q est divisible par 2i et lʼindice de Witt de la forme q au-dessus du corps des fonctions de Q est égal à i, alors la variété Q est 2-incompressible.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.10.004
Nikita A. Karpenko 1

1 UPMC Sorbonne universités, institut de mathématiques de Jussieu, 4, place Jussieu, 75252 Paris cedex 05, France
@article{CRMATH_2011__349_21-22_1131_0,
     author = {Nikita A. Karpenko},
     title = {Incompressibility of orthogonal grassmannians},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1131--1134},
     publisher = {Elsevier},
     volume = {349},
     number = {21-22},
     year = {2011},
     doi = {10.1016/j.crma.2011.10.004},
     language = {en},
}
TY  - JOUR
AU  - Nikita A. Karpenko
TI  - Incompressibility of orthogonal grassmannians
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 1131
EP  - 1134
VL  - 349
IS  - 21-22
PB  - Elsevier
DO  - 10.1016/j.crma.2011.10.004
LA  - en
ID  - CRMATH_2011__349_21-22_1131_0
ER  - 
%0 Journal Article
%A Nikita A. Karpenko
%T Incompressibility of orthogonal grassmannians
%J Comptes Rendus. Mathématique
%D 2011
%P 1131-1134
%V 349
%N 21-22
%I Elsevier
%R 10.1016/j.crma.2011.10.004
%G en
%F CRMATH_2011__349_21-22_1131_0
Nikita A. Karpenko. Incompressibility of orthogonal grassmannians. Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1131-1134. doi : 10.1016/j.crma.2011.10.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.004/

[1] P. Brosnan On motivic decompositions arising from the method of Białynicki-Birula, Invent. Math., Volume 161 (2005) no. 1, pp. 91-111

[2] V. Chernousov; A. Merkurjev Motivic decomposition of projective homogeneous varieties and the Krull–Schmidt theorem, Transform. Groups, Volume 11 (2006) no. 3, pp. 371-386

[3] R. Elman; N. Karpenko; A. Merkurjev The Algebraic and Geometric Theory of Quadratic Forms, American Mathematical Society Colloquium Publications, vol. 56, American Mathematical Society, Providence, RI, 2008

[4] N. Karpenko; A. Merkurjev Essential dimension of quadrics, Invent. Math., Volume 153 (2003) no. 2, pp. 361-372

[5] N.A. Karpenko, Canonical dimension, in: Proceedings of the ICM 2010, vol. II, pp. 146–161.

[6] N.A. Karpenko Incompressibility of generic orthogonal grassmannians, Linear Algebraic Groups and Related Structures (preprint server), Volume 409 (2010) (7 pp)

[7] N.A. Karpenko, Upper motives of algebraic groups and incompressibility of Severi–Brauer varieties, Linear Algebraic Groups and Related Structures (preprint server) 333 (2009) 18 pp.; J. Reine Angew. Math., in press.

[8] N.A. Karpenko Cohomology of relative cellular spaces and of isotropic flag varieties, Algebra i Analiz, Volume 12 (2000) no. 1, pp. 3-69

[9] N.A. Karpenko Hyperbolicity of orthogonal involutions, Doc. Math. Extra Volume: Andrei A. Suslinʼs Sixtieth Birthday (2010), pp. 371-392 (electronic). With an Appendix by Jean-Pierre Tignol

[10] N.A. Karpenko Upper motives of outer algebraic groups, Quadratic Forms, Linear Algebraic Groups, and Cohomology, Dev. Math., vol. 18, Springer, New York, 2010, pp. 249-258

[11] N.A. Karpenko; A.S. Merkurjev Canonical p-dimension of algebraic groups, Adv. Math., Volume 205 (2006) no. 2, pp. 410-433

[12] M.-A. Knus; A. Merkurjev; M. Rost; J.-P. Tignol The Book of Involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998 (With a preface in French by J. Tits)

[13] B.G. Mathews, Canonical dimension of projective homogeneous varieties of inner type A and type B, ProQuest LLC, PhD Thesis, University of California, Los Angeles, Ann Arbor, MI, 2009.

[14] A. Vishik, Direct summands in the motives of quadrics, preprint, 1999, 13 pp. Available on the web page of the author.

[15] A. Vishik Fields of u-invariant 2r+1, Algebra, Arithmetic, and Geometry: In Honor of Yu.I. Manin, vol. II, Progr. Math., vol. 270, Birkhäuser Boston Inc., Boston, MA, 2009, pp. 661-685

Cited by Sources:

Comments - Policy


Articles of potential interest

Classification of upper motives of algebraic groups of inner type An

Charles De Clercq

C. R. Math (2011)


Motivic decompositions of projective homogeneous varieties and change of coefficients

Charles De Clercq

C. R. Math (2010)


Motivic decomposability of generalized Severi–Brauer varieties

Maksim Zhykhovich

C. R. Math (2010)