47 research outputs found
Special correspondences and Chow traces of Landweber-Novikov operations
We prove that the function field of a variety which possesses a special
correspondence in the sense of M. Rost preserves the rationality of cycles of
small codimensions. This fact was proven by Vishik in the case of quadrics and
played the crucial role in his construction of fields with -invariant
. The main technical tools are algebraic cobordism of Levine-Morel,
generalized Rost degree formula and divisibility of Chow traces of certain
Landweber-Novikov operations. As a direct application of our methods we prove
the Vishik's Theorem for all -varieties.Comment: 13 page
Towards generalized cohomology Schubert calculus via formal root polynomials
An important combinatorial result in equivariant cohomology and -theory
Schubert calculus is represented by the formulas of Billey and Graham-Willems
for the localization of Schubert classes at torus fixed points. These formulas
work uniformly in all Lie types, and are based on the concept of a root
polynomial. In this paper we define formal root polynomials associated with an
arbitrary formal group law (and thus a generalized cohomology theory). We focus
on the case of the hyperbolic formal group law (corresponding to elliptic
cohomology). We study some of the properties of formal root polynomials. We
give applications to the efficient computation of the transition matrix between
two natural bases of the formal Demazure algebra in the hyperbolic case. As a
corollary, we rederive in a transparent and uniform manner the formulas of
Billey and Graham-Willems. We also prove the corresponding formula in
connective -theory, which seems new, and a duality result in this case.
Other applications, including some related to the computation of Bott-Samelson
classes in elliptic cohomology, are also discussed.Comment: 20pp, several minor typos and mistakes were corrected; references
updated; the analogue of the Billey-Graham-Willems formula and the respective
duality result for connective K-theory was adde
The gamma-filtration and the Rost invariant
Let X be the variety of Borel subgroups of a simple and strongly inner linear
algebraic group G over a field k. We prove that the torsion part of the second
quotient of Grothendieck's gamma-filtration on X is a cyclic group of order the
Dynkin index of G. As a byproduct of the proof we obtain an explicit cycle that
generates this cyclic group; we provide an upper bound for the torsion of the
Chow group of codimension-3 cycles on X; we relate the generating cycle with
the Rost invariant and the torsion of the respective generalized Rost motives;
we use this cycle to obtain a uniform lower bound for the essential dimension
of (almost) all simple linear algebraic groups.Comment: 19 pages; this is an essentially extended version of the previous
preprint. Applications to cohomological invariants and essential dimensions
of linear algebraic groups are provide
J-invariant of linear algebraic groups
Let G be a linear algebraic group over a field F and X be a projective
homogeneous G-variety such that G splits over the function field of X. In the
present paper we introduce an invariant of G called J-invariant which
characterizes the motivic behaviour of X. This generalizes the respective
notion invented by A. Vishik in the context of quadratic forms. As a main
application we obtain a uniform proof of all known motivic decompositions of
generically split projective homogeneous varieties (Severi-Brauer varieties,
Pfister quadrics, maximal orthogonal Grassmannians, G2- and F4-varieties) as
well as provide new examples (exceptional varieties of types E6, E7 and E8). We
also discuss relations with torsion indices, canonical dimensions and
cohomological invariants of the group G.Comment: The paper containes 39 pages and uses XYPIC packag
Invariants of degree 3 and torsion in the Chow group of a versal flag
We prove that the group of normalized cohomological invariants of degree 3
modulo the subgroup of semidecomposable invariants of a semisimple split linear
algebraic group G is isomorphic to the torsion part of the Chow group of
codimension 2 cycles of the respective versal G-flag. In particular, if G is
simple, we show that this factor group is isomorphic to the group of
indecomposable invariants of G. As an application, we construct nontrivial
cohomological classes for indecomposable central simple algebras.Comment: Appendix with computations for the PGO8-case is adde