
On special Clifford groups and their characteristic classes
Preprint (20 Apr 2023, 18 pages).

On characteristic classes modulo torsion for spin groups
Preprint (revised version of 23 Mar 2023, 13 pages).

Partially splitting PGOtorsors
Preprint (3 Mar 2023, 17 pages).

Notes on spin grassmannians
Preprint (20 Oct 2022, 7 pages).
Not intended for a journal submission.

Chow ring of BSO(2n) in characteristic 2
Final preprint version (19 Oct 2022, 11 pages).
To appear in
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5).

Filtrations on the representation ring of an affine algebraic group
Final preprint version (21 Dec 2022, 14 pages).
To appear in
Q. J. Math.

Envelopes and classifying spaces
Final preprint version (11 Dec 2022, 10 pages).
To appear in Math. Nachr.

On classifying spaces of spin groups
Results Math.
77 (2022), no. 4, paper no. 144, 8 pages.
Final preprint version (14 May 2022, 7 pages).
Correction:
in the bottom line of the commutative square preceding Remark 6
the following letters should be removed: "f" and both appearances of "W".
Erratum: see Appendix in
On characteristic classes modulo torsion for spin groups
posted above.

(with A. Merkurjev)
Poincaré duality for tautological Chern subrings of orthogonal grassmannians
Math. Scand.
128 (2022), 221228.
Final preprint version (6 Dec 2021, 7 pages).
CAUTION: numbering of statements in the journal version differs from the preprint.

(with S. Baek)
Yagita's counterexamples and beyond
To appear in
Arkiv för Matematik.
Final reprint version (16 Jun 2022, 13 pages).

(with A. Merkurjev)
Indexes of generic grassmannians for spin groups
Proc. Lond. Math. Soc. (3)
125 (2022), no. 4, 825840.
Final preprint version (7 May 2022, 15 pages).

On generic flag varieties for odd spin groups
To appear in
Publicacions Matemàtiques.
Final preprint version (23 Nov 2021, 11 pages).

(with R. A. Devyatov and A. S. Merkurjev)
Maximal indexes of flag varieties for spin groups
Forum of Mathematics, Sigma
9 (2021): e34, 112.
doi:10.1017/fms.2021.30

(with A. S. Merkurjev)
The Chow ring of the classifying space of the unitary group
Algebra and Number Theory
16 (2022), no. 1, 139154.
Final preprint version (12 Feb 2021, 15 pages).

(with A. S. Merkurjev)
Equivariant connective Ktheory
J. Algebraic Geom.
31 (2022), 181204.
Final preprint version (30 Jul 2020, 19 pages).

An ultimate proof of HoffmannTotaro's conjecture
To appear in
J. Eur. Math. Soc. (JEMS).
Final preprint version (14 Dec 2021, 13 pages).

Extended PetrovSemenov's connections
Archiv der Mathematik 115 (2020), 2734.
doi:
10.1007/s00013020014383
Final preprint version
(29 Dec 2019, 7 pages).

A remark on connective Ktheory
J. Algebra
560 (2020), 12111218.
Final preprint version (4 June 2020, 7 pages).

A counterexample by Yagita
International Journal of Mathematics
31 (2020) no. 3, 2050025 (10 pages).
doi:
10.1142/S0129167X20500251
Final preprint version
(26 Dec 2019, 9 pages).

(with A. S. Merkurjev)
Chow filtration on representation rings of algebraic groups
International Mathematics Research Notices 9 (2021), 66916716.
doi:
10.1093/imrn/rnz049
Final preprint version
(5 February 2018, 20 pages).
CAUTION: numbering of statements in the journal version differs from the preprint.

(with E. Mackall and with appendices by E. Mackall)
On the Ktheory coniveau epimorphism for products of SeveriBrauer varieties
Annals of KTheory 4 (2019), no. 4, 317344.
Final preprint version
(19 October 2018, 21 pages).

On generic flag varieties of Spin(11) and Spin(12)
Manuscripta Math.
157 (2018), 1321.
Final preprint version
(28 November 2017, 9 pages).
Corrected final preprint version
(21 March 2018, 9 pages).
List of corrections

On generic quadratic forms
Screen.
Print.
Pacific J. Math. 297 (2018), no. 2, 367380.
Correction: [V] should be added in the beginning of the formula in Line 4 of Proof of Theorem 2.1.
Final preprint version
of 29 March 2018, 12 pages.

On generically split generic flag varieties
Bull. Lond. Math. Soc. 50 (2018), 496508.
Final preprint version, of 10 April 2018, 15 pages.

Chow ring of generic flag varieties
Math. Nachr.
290 (2017), 26412647.
Final preprint version (14 February 2017, 9 pages).
CAUTION: numbering of subsections in the journal version differs from the preprint.

Chow ring of generically twisted varieties of complete flags
Adv. Math.
306 (2017), 789806.
Final preprint version
(11 August 2016, 15 pages).

Chow groups of some generically twisted flag varieties
Annals of KTheory
2 (2017), no. 2, 341356.
Final preprint version
(26 April 2016, 13 pages).

Around 16dimensional quadratic forms in I_{q}^{3}
Math. Z. 285 (2017), no. 1, 433444.
Final preprint version (4 June 2016, 13 pages).

Incompressibility of products
of pseudohomogeneous varieties
Canad. Math. Bull.
59 (2016), 824833.
Final preprint version
(20 April 2016, 10 pages).

Incompressibility of products
Revised version
(with removed ``geometrically split'' condition),
8 February 2015, 6 pages.
(Not intended for a journal submission because contained in
Incompressibility of products
of pseudohomogeneous varieties.
)
The version
Linear Algebraic Groups and Related Structures
(preprint server) 547 (2015, Jan 31, 7 pages)
is now obsolete.

Incompressible Grassmannians of isotropic subspaces
Bull. Soc. Math. France 144 (2016), no. 4, 763773.
Final preprint version
(11 February 2016, 9 pages).

Incompressibility of products by Grassmannians of isotropic subspaces
Revised version,
1 February 2015, 12 pages.
(Not intended for a journal submission because
the main result has been covered by
Incompressibility of products of pseudohomogeneous varieties
and the rest is contained in
Incompressible Grassmannians of isotropic subspaces.
)
The version
Linear Algebraic Groups and Related Structures
(preprint server) 541 (2014, Sep 23, 12 pages)
is now obsolete.

Unitary Grassmannians of division algebras
Transform. Groups 21 (2016), no. 1, pp. 115127.
Final preprint version
(31 December 2014, 12 pages).
The version
Linear Algebraic Groups and Related Structures
(preprint server) 539 (2014, August 22, 9 pages).
is now obsolete.

Incompressibility of products of Weil transfers of generalized SeveriBrauer varieties
Math. Z.
279 (2015), no. 34, 767777.
Final preprint version.
The version
Linear Algebraic Groups and Related Structures
(preprint server) 536 (2014, June 26, 12 pages)
is now obsolete.

(with Z. Reichstein and with an appendix by J. Pevtsova and Z. Reichstein)
A numerical invariant for linear representations of finite groups
Comment. Math. Helv.
90 (2015), no. 3, 667–701.
Final preprint version.
The version
Linear Algebraic Groups and Related Structures
(preprint server) 534 (extended version 2014, June 18, 24 pages)
is now obsolete.

(with A. S. Merkurjev)
Motivic decomposition of compactifications
of certain group varieties
J. Reine Angew. Math. 745 (2018), 4158.
Final preprint version
(26 Feb 2016, 18 pages).
The version
Linear Algebraic Groups and Related Structures
(preprint server) 528 (extended version 2014, Feb 22, 18 pages)
is now obsolete.

Minimal canonical dimensions of quadratic forms
Documenta Math. Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015),
367385.
Final preprint version
(February 2014, 17 pages).
The previous version
Linear Algebraic Groups and Related Structures
(preprint server) 509 (2013, Sep 1, revised 2013, Sep 13)
is now obsolete.

(with A. S. Merkurjev)
Hermitian forms over quaternion algebras
Compos. Math.
150 (2014), 20732094.
Final preprint version
(May 2014, 24 pages).
The previous version
Linear Algebraic Groups and Related Structures
(preprint server) 498 (2013, May 13, 22 pages)
is now obsolete.

Incompressibility of generic torsors of norm tori
Zap. Nauchn. Sem. S.Peterburg. Otdel. Mat. Inst. Steklov. (POMI)
414
(2013), 106112.
Final preprint version
(2013, Jan 25, 6 pages).
The initial version
Linear Algebraic Groups and Related Structures
(preprint server) 475 (2012, Aug 8, 5 pages)
is now obsolete.

Orthogonal and symplectic grassmannians of division algebras
J. Ramanujan Math. Soc. 28 (2013), no. 2, 213222.
Final preprint version
(2012, Dec 3, 9 pages).
The version
Linear Algebraic Groups and Related Structures
(preprint server) 469 (2012, May 21, 8 pages)
is obsolete.

Sufficiently generic orthogonal grassmannians
J. Algebra 372 (2012) 365375.
Final preprint version
(2012, Oct 8, 13 pages).

(with A. S. Merkurjev)
On standard norm varieties
Ann. Sci. Éc. Norm. Supér. (4)
46, fascicule 1 (2013), 175214.
Final preprint version
(18 June 2012, 38 pages).
The version
Linear Algebraic Groups and Related Structures
(preprint server) 456 (2012, Jan 4, 37 pages)
is obsolete.

Unitary grassmannians
J. Pure Appl. Algebra
216 (2012), no. 12, 25862600
Final preprint version
(15 Feb. 2012, 25 pages).
Here is a
corrected version
of Example 9.12.
The version
Linear Algebraic Groups and Related Structures
(preprint server) 424 (2011, Mar 17, revised 2011, Oct 9, 22 pages)
is obsolete.

Variations on a theme of rationality of cycles
Cent. Eur. J. Math.
11 (2013), no. 6, 10561067.
Final preprint version
(16 July 2012, 14 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. Math. Acad. Sci. Paris 349 (2011) 11311134.
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
Acta Math.
211 (2013), no. 2, 227253.
Final preprint version
(4 June 2012, 22 pages).
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 a journal submission because contained in
Sufficiently generic orthogonal grassmannians.
)

Hyperbolicity of unitary involutions
Sci. China Math.
55 (2012), no. 5, 937945.
The original publication is available at www.scichina.com and www.springerlink.com
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 SeveriBrauer variety.
Note, Jul 2010, 4 pages.

Isotropy of symplectic involutions
C. R. Math. Acad. Sci. Paris. 348 (2010), no. 2122, 11511153.
Final preprint version.

Canonical dimension
Proceedings of the ICM 2010, vol. II, 146161.

(with an appendix by J.P. Tignol)
Isotropy of orthogonal involutions
Amer. J. Math. 135 (2013), 115.
Final preprint version
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 SeveriBrauer varieties
J. Inst. Math. Jussieu
11 (2012), no. 01, 119131.

Upper motives of outer algebraic groups
In Quadratic forms, linear algebraic
groups, and cohomology, vol. 18 of Dev. Math. Springer, New York, 2010, pp. 249258.
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 SeveriBrauer varieties
J. Reine Angew. Math. 677 (2013), 179198.
Final preprint version.
CAUTION: numbering of subsections in the journal version differs from the preprint.
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 JeanPierre Tignol
Hyperbolicity of orthogonal involutions
Doc. Math. Extra Volume: Andrei A. Suslin's Sixtieth Birthday (2010),
371392.
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 a journal submission 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.
ISBN10: 082184329X, ISBN13: 9780821843291.
List of corrections

On isotropy of quadratic pair
Contemp. Math. 493 (2009), 211218.
Preprint version

(with A. S. Merkurjev)
Essential dimension of finite pgroups.
Invent. Math.
172, no. 3, 491508 (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. ColliotThélène and A. Merkurjev)
Rational surfaces and canonical dimension of PGL_6
St. Petersburg Math. J.
19 (2008), no. 5, 793804.
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 (JeanPierre Serre special issue, part I), 10331039.
An older version:
Preprint of the MaxPlanckInstitut für Mathematik
MPIM2005109
(December 2005), 5 pages.

(with A. S. Merkurjev)
Canonical pdimension of algebraic groups.
Adv. Math.
205 (2006), 410433.
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, 391404.
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, 522529.
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, 211215.
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, 7786,
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 a journal submission).

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, 9731002.
An obsolete preprint version:
Linear Algebraic Groups and Related Structures
(preprint server) 128 (December 2003), 129.

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. 1118, 2002, Universidad de Talca, Chile.
Contemp. Math. 344, 207213.
Preprint version:
Prépublications de l'IHES
M/03/08 (February 2003), 17.

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), 131.

(with A. S. Merkurjev)
Rost projectors and Steenrod operations.
Doc. Math. 7 (2002), 481493.

On the first Witt index of quadratic forms.
Invent. Math.
153 (2003), no. 2, 455462.
Preprint version (1 December 2002)
,
An older version:
Linear Algebraic Groups and Related Structures
(preprint server) 91 (June 2002), 17.

(with A. S. Merkurjev)
Essential dimension of quadrics.
Invent. Math.
153 (2003), no. 2, 361372.
Preprint version (December 2002)
An older version:
Linear Algebraic Groups and Related Structures
(preprint server) 90 (June 2002), 19.

Motives and Chow groups of quadrics
with application
to the uinvariant
(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), 195227.
Preprint version (September 1999), 133.

On anisotropy of orthogonal involutions.
J. Ramanujan Math. Soc.
15 (2000), no. 1, 122.
Preprint version
(October 1999),
115.

Weil transfer of algebraic cycles.
Indag. Math. (N.S.)
11 (2000), no. 1, 7386.
Preprint version (August 18, 1999).

Criteria of motivic equivalence for quadratic forms
and central simple algebras.
Math. Ann.
317 (2000), no. 3, 585611.
Preprint version (October, 1999)

Cohomology of relative cellular spaces
and isotropic flag varieties.
St. Petersburg Math. J.
12 (2001), no. 1, 150.
Preprint version.
The russian version
of this text
(appeared in
Algebra i Analiz 12 (2000), no. 1, 369).

A shortened construction of the Rost motive.
Preprint (November, 1998), 14.
This text is a modified version of:
A direct construction of the Rost motive.
KTheory 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 nonexcellence.
Algebras and Representation Theory
2 (1999), no. 1,
1959.
Preprint version.

(with Oleg Izhboldin)
Some new examples in the theory of quadratic forms.
Math. Z.
234
(2000) 4,
647695.

Three theorems on common splitting fields
of central simple algebras.
Israel J. Math.
111 (1999), 125141.
Preprint version.

Cohomological Invariants
of Homogeneous Varieties
(with Applications to
Quadratic Forms
and Central Simple Algebras).
Habilitationsschrift (1997), 1172.

(with Oleg Izhboldin)
Isotropy of 8dimensional quadratic forms
over function fields of quadrics.
Comm. Algebra
27 (1999), no. 4,
18231841.

(with Oleg Izhboldin)
On the group H^{3}(F(\psi,D)/F).
Doc. Math. 2 (1997),
297311.

(with Anne Quéguiner)
A criterion of decomposability for degree 4
algebras with unitary involution.
J. Pure Appl. Algebra
147 (2000),
no. 3,
303309.
Preprint version.

(with Oleg Izhboldin)
Isotropy of 6dimensional quadratic forms over function fields
of quadrics.
J. Algebra
209 (1998), 6593.
Preprint version.

(with Oleg Izhboldin)
Isotropy of virtual Albert forms over function fields
of quadrics.
Math. Nachr.
206 (1999), 111122.

Cycles de codimension 2 sur produits de
variétés de SeveriBrauer.
Publ. Math. Fac. Sci. Besançon,
Théorie des Nombres, Années 1994/951995/96,
115.

Order of torsion in CH^{4} of quadrics.
Doc. Math. 1 (1996),
5765.

Codimension 2 cycles on SeveriBrauer varieties.
KTheory
13 (1998), no. 4,
305330.

Torsion in CH^{2} of SeveriBrauer varieties
and indecomposability of generic algebras.
Manuscripta Math.
88 (1995), 109117.
Preprint version.

On topological filtration for triquaternion algebra.
Nova J. Math. Game Theory Algebra
4
(1996), no. 2, 161167.

The Grothendieck Chowmotifs of SeveriBrauer varieties.
St. Petersbourg Math. J.
7 (1996), no. 4, 649661.
Preprint version.

Chow groups of quadrics and index reduction formula.
Nova J. Algebra Geom.
3 (1995), no. 4, 357379.

On topological filtration for SeveriBrauer varieties II.
Amer. Math. Soc. Transl. 174 (1996), no. 2, 4548.
Preprint version.

On topological filtration for SeveriBrauer varieties.
Proc. Symp. Pure Math. 58.2 (1995), 275277.
Preprint version.

Chow groups of quadrics and the stabilization conjecture.
Adv. Soviet Math. 4 (1991), 38.

Cycles of codimension 3 on a projective quadric.
J. Sov. Math.
63 (1993), no. 6, 678682.

The Grothendieck ring of quadrics, and gamma filtration. (Russian)
Rings and modules. Limit theorems of probability theory, No. 3 (Russian),
39–61, 256, Izd. St.Peterbg. Univ., St. Petersburg, 1993.
Preprint version (Russian, with a preliminary title)
Abstract (Russian)
A
recommendation for publication (Russian)

The Chow ring of a projective quadric
(Russian)
PhD Thesis, Leningrad, USSR, June 1990, 82 pages.

(with A. S. Merkurjev)
Chow groups of projective quadrics.
Leningrad (St. Petersburg) Math. J.
2 (1991), no. 3, 655671.

Algebrogeometric invariants of quadratic forms.
Leningrad (St. Petersburg) Math. J.
2 (1991), no. 1, 119138.
Russian original
(Algebra i Analiz 2 (1990), vyp. 1).