77 research outputs found
Unitary grassmannians
We study projective homogeneous varieties under an action of a projective
unitary group (of outer type). We are especially interested in the case of
(unitary) grassmannians of totally isotropic subspaces of a hermitian form over
a field, the main result saying that these grassmannians are 2-incompressible
if the hermitian form is generic. Applications to orthogonal grassmannians are
provided.Comment: 25 page
Holes in I^n
Let F be an arbitrary field of characteristic not 2. We write W(F) for the
Witt ring of F, consisting of the isomorphism classes of all anisotropic
quadratic forms over F. For any element x of W(F), dimension dim x is defined
as the dimension of a quadratic form representing x. The elements of all even
dimensions form an ideal denoted I(F). The filtration of the ring W(F) by the
powers I(F)^n of this ideal plays a fundamental role in the algebraic theory of
quadratic forms. The Milnor conjectures, recently proved by Voevodsky and
Orlov-Vishik-Voevodsky, describe the successive quotients I(F)^n/I(F)^{n+1} of
this filtration, identifying them with Galois cohomology groups and with the
Milnor K-groups modulo 2 of the field F. In the present article we give a
complete answer to a different old-standing question concerning I(F)^n, asking
about the possible values of dim x for x in I(F)^n. More precisely, for any
positive integer n, we prove that the set dim I^n of all dim x for all x in
I(F)^n and all F consisists of 2^{n+1}-2^i, i=1,2,...,n+1 together with all
even integers greater or equal to 2^{n+1}. Previously available partial
informations on dim I^n include the classical Arason-Pfister theorem, saying
that no integer between 0 and 2^n lies in dim I^n, as well as a recent Vishik's
theorem, saying the same on the integers between 2^n and 2^n+2^{n-1} (the case
n=3 is due to Pfister, n=4 to Hoffmann). Our proof is based on computations in
Chow groups of powers of projective quadrics (involving the Steenrod
operations); the method developed can be also applied to other types of
algebraic varieties.Comment: 29 page
Incompressibility of orthogonal grassmannians
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 in the interval
[1, (\dim q)/2]). If the degree of each closed point on Q is divisible by 2^i
and the Witt index of q over the function field of Q is equal to i, then the
variety Q is 2-incompressible.Comment: 5 page
On standard norm varieties
Let be a prime integer and a field of characteristic 0. Let be
the {\em norm variety} of a symbol in the Galois cohomology group
(for some ), constructed in the proof of
the Bloch-Kato conjecture. The main result of the paper affirms that the
function field has the following property: for any equidimensional
variety , the change of field homomorphism \CH(Y)\to\CH(Y_{F(X)}) of Chow
groups with coefficients in integers localized at is surjective in
codimensions . 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 {\em -triviality} of , the property saying that the degree
homomorphism on \CH_0(X_L) is injective for any field extension with
. The proof involves the theory of rational correspondences
reviewed in Appendix.Comment: 38 pages; final version, to appear in Ann. Sci. \'Ec. Norm. Sup\'er.
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On generic flag varieties for odd spin groups
Altres ajuts: The author's work has been supported by a Discovery Grant from the National Science and Engineering Research Council of Canada.For the spin group G = Spin2n+1 with arbitrary n, a generic G-torsor E over a field, and a parabolic subgroup P ⊂ G, we consider the generic flag variety E/P and describe its Chow ring modulo torsion. This description determines the index of E/P, completing results of [3], where the index has been determined for most P
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