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Calculs explicites dans les groupes de Grotendieck et de Chow des variétés homogènes projectives

Abstract : The homogeneous projective varieties under a split algebraic group have
a rather simple geometry. The Bruhat's decomposition gives a cellular
decomposition of theses varieties. Therefore the Chow ring of such
varieties has a basis consisting of the classes of the closure of
these cells, the Schubert varieties. The same thing is true for the
Grothendieck ring of such varieties. This implies that these two rings are
torsion-free. More precisely, the preceding basis of the Grothendieck
ring provides the topological filtration of this ring and therefore
gives once again, via the graded ring associated, the basis of the
Chow ring. On the other hand, there exists a second basis found by
Pittie and Steinberg of the Grothendieck ring of these varieties;
basis which is invariant under the action of the Galois group.

The chapter II of the dissertation deals about known results about the
combinatorics which give the expression of the structural sheaves of
the Schubert varities in the Grothendieck ring, in the case of
complete flag varieties associated to a vector space. It allows,
following in particular works by Lascoux, to give a combinatorial
expression of the basis change matrix between the two above basis. In
the case of complete flag varieties in a vector space of dimension
three, we give explicit resolutions of the structural sheaves of the
Schubert varieties by mean of fibers of Pittie's basis.

The Chow groups are well-known in codimension one and have been
studied by Karpenko in codimension two in the case of Severi-Brauer
varieties. The computation of motives of homogeneous projective
varieties under the projective linear group associated to a simple
central algebra over a field can be reduced, under some hypothesis to
the computation of motives of generalized Severi-Brauer varieties,
forms of grassmanians, as Calmès, Petrov, Semenov and Zainouline have
shown. In chapter III, we construct explicit isomorphisms of
varieties which allow us to reduce the computation of the Chow groups
of these projective homogenous varieties to the computation of Chow
groups of generalized Severi-Brauer varieties.

The material described in the chapter III is used one more time in
chapter IV to give a new proof of a result by Karpenko on the
decomposition of the motive of the Severi-Brauer variety associated to
a matrix algebra with coefficients in a simple central algebra.
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Submitted on : Tuesday, December 19, 2006 - 9:09:30 AM
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Franck Doray. Calculs explicites dans les groupes de Grotendieck et de Chow des variétés homogènes projectives. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2006. Français. ⟨tel-00120949⟩

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