On motivic decompositions arising from the method of Białynicki-Birula

Abstract

Recently, V. Chernousov, S. Gille and A. Merkurjev have obtained a decomposition of the motive of an isotropic smooth projective homogeneous variety analogous to the Bruhat decomposition. Using the method of A. Bialynicki-Birula and a corollary, which is essentially due to S. del Baño, I generalize this decomposition to the case of a (possibly anisotropic) smooth projective variety homogeneous under the action of an isotropic reductive group. This answers a question of N. Karpenko.

Equivariant K-theory and higher Chow groups of schemes

“…Remark We remark that the non‐equivariant version of Proposition is known to exist in the literature in various forms. For instance, in view of the comparison of Chow motives and Voevodsky's derived category of motives and the relation between the motivic cohomology and higher Chow groups, the non‐equivariant version follows from the results of Brosnan . The results of Brosnan build on the earlier work of Karpenko , generalized by Chernousov–Gille–Merkurjev (see also ).…”

Oriented cohomology and motivic decompositions of relative cellular spaces

“…Among examples of relative cellular spaces are isotropic projective homogeneous varieties H = G/P , where G is a linear algebraic group (not necessarily split) and P a parabolic subgroup (see [3,Theorem 7.5, Proposition 6.3]), and, more generally, smooth projective varieties equipped with an action of the multiplicative group G m (see [1,Theorem 3.2]). …”

“…Note that the shift of degree by −c i in (1) holds under the assumption that A is Z-graded and push-forwards in A raise the degree by the codimension of the morphism. If it is twice the codimension, e.g.…”

“…If it is twice the codimension, e.g. for the singular cohomology of complex varieties, then we get −2c i in (1). If A is bigraded, which is the case for many theories, then the formula must also be changed accordingly, e.g.…”

Canonical Dimension

“…Then the variety X := SB (deg D)/2 (D) is 2-incompressible. Indeed, according to [4] or [5] or [14], the motive M (X) F (X) is a sum of one F 2 , one F 2 (dim X), and of shifts of M (Y ), where Y runs over some projective homogeneous F (X)-varieties with v 2 (Y ) > 0. It follows that U (X) F (X) contains the summand F 2 (dim X).…”