-
(with A. S. Merkurjev)
On standard norm varieties
Preprint, 4 Jan. 2012, 37 pages.
Abstract.
Let p be a prime integer and F a field of characteristic 0.
Let X be the norm variety of a symbol in
the Galois cohomology group Hn+1(F,μp⊗n) (for some n≥1), constructed in the proof of the
Bloch-Kato conjecture.
The main result of the paper affirms
that the function field F(X) has the following property:
for any equidimensional variety Y, the change
of field homomorphism CH(Y)→CH(YF(X))
of Chow groups with coefficients in integers localized at p
is surjective in codimensions < (dim X)/(p-1).
One of the main ingredients of the proof is a computation of Chow groups
of a (generalized) Rost motive (a variant of the main result not relying on this is given in
Appendix).
Another important ingredient is A-triviality of X, the property saying that
the degree homomorphism on CH0(XL) is injective for any field extension L/F with
X(L)≠∅.
The proof involves
the theory of rational correspondences, due to Markus Rost, reviewed in Appendix.
-
Sufficiently generic orthogonal grassmannians.
Preprint, 30 Dec. 2011, 12 pages.
-
Unitary grassmannians
Linear Algebraic Groups and Related Structures
(preprint server) 424 (2011, Mar 17, revised 2011, Oct 9, 22 pages)
-
Variations on a theme of rationality of cycles
Preprint,
revised version
(6 Jan 2012, 13 pages).
The version
Linear Algebraic Groups and Related Structures
(preprint server) 443 (2011, Sep 21, 13 pages)
is obsolete.
-
Incompressibility of orthogonal grassmannians
C. R. Acad. Sci. Paris, Ser. I 349 (2011) 1131--1134.
Final preprint version (July 2011, 5 pages):
arXiv copy
The version
Linear Algebraic Groups and Related Structures
(preprint server) 437 (2011, Jun 30, 5 pages)
is obsolete.
-
(with M. Zhykhovich)
Isotropy of unitary involutions
arXiv version
(May 2011)
The version
Linear Algebraic Groups and Related Structures
(preprint server) 423 (2011, Mar 15, 20 pages)
is obsolete.
-
Incompressibility of generic orthogonal grassmannians
Linear Algebraic Groups and Related Structures
(preprint server) 409 (2010, Nov 23, 7 pages)
(Not intended for publication because contained in
Sufficiently generic orthogonal grassmannians.
)
-
Hyperbolicity of unitary involutions
Sci. China Math.,
to appear.
Final preprint version.
The version
Linear Algebraic Groups and Related Structures
(preprint server) 397 (2010, Jul 30, 10 pages)
is obsolete.
-
Codimension 2 cycles
on quadratic Weil transfer
of biquaternionic Severi-Brauer variety.
Note, Jul 2010, 4 pages.
-
Isotropy of symplectic involutions
C. R. Math. Acad. Sci. Paris. 348 (2010), no. 21-22, 1151--1153.
Final preprint version.
-
Canonical dimension
Proceedings of the ICM 2010, vol. II, 146--161.
-
Isotropy of orthogonal involutions
Revised version, with an Appendix by J.-P. Tignol,
to appear in Amer. J. Math.
The preprint version
Linear Algebraic Groups and Related Structures
(preprint server) 371 (2009, Nov 21, 9 pages).
is obsolete.
-
Incompressibility of quadratic Weil transfer of generalized Severi-Brauer varieties
J. Inst. Math. Jussieu
11 (2012), no. 01, 119-131.
-
Upper motives of outer algebraic groups
In Quadratic forms, linear algebraic
groups, and cohomology, vol. 18 of Dev. Math. Springer, New York, 2010, pp. 249--258.
The preprint version
Linear Algebraic Groups and Related Structures
(preprint server) 340 (2009, Jun 16, 7 pages)
is obsolete
-
Upper motives of algebraic groups and
incompressibility of Severi-Brauer varieties
Final version,
to appear in J. Reine Angew. Math.
The preprint version
Linear Algebraic Groups and Related Structures
(preprint server) 333 (2009, Apr 3, revised: 2009, Apr 24, 18 pages)
is obsolete.
-
with an Appendix by Jean-Pierre Tignol
Hyperbolicity of orthogonal involutions
Doc. Math. Extra Volume: Andrei A. Suslin's Sixtieth Birthday (2010),
371--392.
Final preprint version.
The preprint version
Linear Algebraic Groups and Related Structures
(preprint server) 330 (2009, Mar 25, revised: 2009, Apr 24, 16 pages)
is obsolete.
-
Hyperbolicity of hermitian forms
over biquaternion algebras
(not intended for publication because contained in Hyperbolicity of orthogonal involutions)
Revised version
(January 2009, 17 pages)
Linear Algebraic Groups and Related Structures
(preprint server) 316 (2008, Dec 16, revised: 2009, Jan 07), 17 p.
-
(with R. Elman and A. Merkurjev)
The Algebraic and Geometric Theory of Quadratic Forms.
pdf , dvi
American Mathematical Society Colloquium Publications, 56.
American Mathematical Society, Providence, RI, 2008. 435 pp.
ISBN-10: 0-8218-4329-X, ISBN-13: 978-0-8218-4329-1.
List of corrections!
-
On isotropy of quadratic pair
Contemp. Math. 493 (2009), 211--218.
Preprint version
-
(with A. S. Merkurjev)
Essential dimension of finite p-groups.
Invent. Math.
172, no. 3, 491--508 (2008).
Final preprint version
(the original publication is available at www.springerlink.com)
The preprint version
Essential dimension of finite groups
Linear Algebraic Groups and Related Structures
(preprint server) 263 (2007, Jun 25), 14 p.
is obsolete.
-
(with J.-L. Colliot-Thélène and A. Merkurjev)
Rational surfaces and canonical dimension of PGL_6
St. Petersburg Math. J.
19 (2008), no. 5, 793--804.
Final preprint version.
The preprint version
Linear Algebraic Groups and Related Structures
(preprint server) 231 (2006, Nov 25), 15 p.
is now obsolete.
-
Canonical dimension of (semi-)spinor groups of small ranks.
Pure Appl. Math. Q.
4 (2008), no. 4 (Jean-Pierre Serre special issue, part I), 1033--1039.
An older version:
Preprint of the Max-Planck-Institut für Mathematik
MPIM2005-109
(December 2005), 5 pages.
-
(with A. S. Merkurjev)
Canonical p-dimension of algebraic groups.
Adv. Math.
205 (2006), 410--433.
The preprint version
Linear Algebraic Groups and Related Structures
(preprint server) 170 (December 2004), 22 p.
is now obsolete.
-
A bound for canonical dimension
of the (semi-)spinor groups.
Duke Math. J.
133 (2006), no. 2, 391--404.
Revised preprint version.
The preprint version
A bound for canonical dimension
of the spinor group,
Linear Algebraic Groups and Related Structures
(preprint server) 171 (December 2004), 10 p.
is now obsolete.
-
(with J. Hurrelbrink and U. Rehmann)
The minimal height of quadratic forms of given dimension.
Arch. Math. (Basel) 87 (2006), no. 6, 522--529.
Preprint version:
Linear Algebraic Groups and Related Structures
(preprint server) 157 (October 2004), 8 p.
-
Canonical dimension of orthogonal groups.
(the pdf file has a wrong pagination)
Transform. Groups 10 (2005), no. 2, 211--215.
An obsolete preprint version:
Linear Algebraic Groups and Related Structures
(preprint server) 144 (July 2004), 4 p.
-
A relation between higher Witt indices.
Revised version
Proceedings of the St. Petersburg Mathematical Society. Vol. XI, 77--86,
Amer. Math. Soc. Transl. Ser. 2, 218,
Amer. Math. Soc., Providence, RI, 2006.
An obsolete preprint version:
Linear Algebraic Groups and Related Structures
(preprint server) 141 (June 2004), 10 p.
-
A descent of motivic isomorphisms.
Preprint, 2 p (not intended for publication).
-
Cycles on powers of quadrics.
Mathematisches Institut, Seminars 2003/04, Universitätsverlag
Göttingen, (2004)
-
Holes in I
n.
Ann. Sci. Ecole Norm. Sup. (4) 37 (2004), no. 6, 973--1002.
An obsolete preprint version:
Linear Algebraic Groups and Related Structures
(preprint server) 128 (December 2003), 1-29.
-
Third proof of second gap in dimensions of quadratic forms from
I n.
Proceedings of the International Conference on the Algebraic and Arithmetic Theory
of Quadratic Forms,
Dec. 11--18, 2002, Universidad de Talca, Chile.
Contemp. Math. 344, 207--213.
Preprint version:
Prépublications de l'IHES
M/03/08 (February 2003), 1-7.
-
Izhboldin's results on stably birational equivalence
of quadrics.
Springer
Lect. Notes Math. 1835
(Proceedings of the Summer School
``Geometric Methods in the Algebraic Theory
of Quadratic Forms'',
Lens, June 2000).
Preprint version:
Linear Algebraic Groups and Related Structures
(preprint server) 96 (July 2002), 1-31.
-
(with A. S. Merkurjev)
Rost projectors and Steenrod operations.
Doc. Math. 7 (2002),
481-493.
-
On the first Witt index of quadratic forms.
Invent. Math.
153 (2003), no. 2, 455-462.
Preprint version (1 December 2002)
,
An older version:
Linear Algebraic Groups and Related Structures
(preprint server) 91 (June 2002), 1-7.
-
(with A. S. Merkurjev)
Essential dimension of quadrics.
Invent. Math.
153 (2003), no. 2, 361-372.
Preprint version (December 2002)
An older version:
Linear Algebraic Groups and Related Structures
(preprint server) 90 (June 2002), 1-9.
-
Motives and Chow groups of quadrics
with application
to the u-invariant
(after Oleg Izhboldin).
Springer
Lect. Notes Math. 1835
(Proceedings of the Summer School
``Geometric Methods in the Algebraic Theory
of Quadratic Forms'',
Lens, June 2000).
Preprint version
-
Characterization of minimal Pfister neighbors via Rost projectors
.
J. Pure Appl. Algebra
160 (2001), 195-227.
Preprint version (September 1999), 1-33.
-
On anisotropy of orthogonal involutions.
J. Ramanujan Math. Soc.
15 (2000), no. 1, 1-22.
Preprint version
(October 1999),
1-15.
-
Weil transfer of algebraic cycles.
Indag. Math. (N.S.)
11 (2000), no. 1, 73-86.
Preprint version (August 18, 1999).
-
Criteria of motivic equivalence for quadratic forms
and central simple algebras.
Math. Ann.
317 (2000), no. 3, 585-611.
Preprint version (October, 1999)
-
Cohomology of relative cellular spaces
and isotropic flag varieties.
St. Petersburg Math. J.
12 (2001), no. 1, 1-50.
Preprint version.
The russian version
of this text
(appeared in
Algebra i Analiz 12 (2000), no. 1, 3-69).
-
A shortened construction of the Rost motive.
Preprint (November, 1998), 1-4.
This text is a modified version of:
A direct construction of the Rost motive.
K-Theory Preprint Archives
282 (1998)
and is dissolved in
Characterization of minimal Pfister neighbors via Rost projectors
(see above).
-
(with Oleg Izhboldin)
Generic splitting fields of
central simple algebras: Galois cohomology and non-excellence.
Algebras and Representation Theory
2 (1999), no. 1,
19-59.
Preprint version.
-
(with Oleg Izhboldin)
Some new examples in the theory of quadratic forms.
Math. Z.
234
(2000) 4,
647-695.
Preprint version (June, 1997).
-
Three theorems on common splitting fields
of central simple algebras.
Israel J. Math.
111 (1999), 125-141.
Preprint version.
-
Cohomological Invariants
of Homogeneous Varieties
(with Applications to
Quadratic Forms
and Central Simple Algebras).
Habilitationsschrift (1997), 1-172.
-
(with Oleg Izhboldin)
Isotropy of 8-dimensional quadratic forms
over function fields of quadrics.
Comm. Algebra
27 (1999), no. 4,
1823--1841.
-
(with Oleg Izhboldin)
On the group H3(F(\psi,D)/F).
Doc. Math. 2 (1997),
297-311.
-
(with Anne Quéguiner)
A criterion of decomposability for degree 4
algebras with unitary involution.
J. Pure Appl. Algebra
147 (2000),
no. 3,
303-309.
Preprint version.
-
(with Oleg Izhboldin)
Isotropy of 6-dimensional quadratic forms over function fields
of quadrics.
J. Algebra
209 (1998), 65-93.
Preprint version.
-
(with Oleg Izhboldin)
Isotropy of virtual Albert forms over function fields
of quadrics.
Math. Nachr.
206 (1999), 111-122.
-
Cycles de codimension 2 sur produits de
variétés de Severi-Brauer.
Publ. Math. Fac. Sci. Besançon,
Théorie des Nombres, Années 1994/95-1995/96,
1-15.
-
Order of torsion in CH4 of quadrics.
Doc. Math. 1 (1996),
57-65.
-
Codimension 2 cycles on Severi-Brauer varieties.
K-Theory
13 (1998), no. 4,
305-330.
Preprint version.
-
Torsion in CH2 of Severi-Brauer varieties
and indecomposability of generic algebras.
Manuscripta Math.
88 (1995), 109-117.
Preprint version.
-
On topological filtration for triquaternion algebra.
Nova J. Math. Game Theory Algebra
4
(1996), no. 2, 161-167.
-
The Grothendieck Chow-motifs of Severi-Brauer varieties.
St. Petersbourg Math. J.
7 (1996), no. 4, 649-661.
Preprint version.
-
Chow groups of quadrics and index reduction formula.
Nova J. Algebra Geom.
3 (1995), no. 4, 357-379.
-
On topological filtration for Severi-Brauer varieties II.
Amer. Math. Soc. Transl. 174 (1996), no. 2, 45-48.
Preprint version.
-
On topological filtration for Severi-Brauer varieties.
Proc. Symp. Pure Math. 58.2 (1995), 275-277.
Preprint version.
-
Chow groups of quadrics and the stabilization conjecture.
Adv. Soviet Math. 4 (1991), 3-8.
-
Cycles of codimension 3 on a projective quadric.
J. Sov. Math.
63 (1993), no. 6, 678-682.
-
(with A. S. Merkurjev)
Chow groups of projective quadrics.
Leningrad (St. Petersburg) Math. J.
2 (1991), no. 3, 655-671.
-
Algebro-geometric invariants of quadratic forms.
Leningrad (St. Petersburg) Math. J.
2 (1991), no. 1, 119-138.
English translation.