Publications of Nikita Karpenko

• 
  (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,μ_p^⊗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(Y_F(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(X_L) 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  is obsolete.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  is obsolete.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  is obsolete.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  is obsolete.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.