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Register a new userChow motives of twisted flag varieties
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Let G be an adjoint simple algebraic group of inner type. We express the Chow motive (with integral coefficients) of some anisotropic projective G-homogeneous varieties in terms of motives of simpler G-homogeneous varieties, namely, those that correspond to maximal parabolic subgroups of G. We decompose the motive of a generalized Severi-Brauer variety SB_2(A), where A is a division algebra of degree 5, into a direct sum of two indecomposable motives. As an application we provide another counter-example to the uniqueness of a direct sum decomposition in the category of motives with integral coefficients.
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We prove that the Chow motives of two smooth cubic fourfolds whose Kuznetsov components are Fourier-Mukai derived-equivalent are isomorphic as Frobenius algebra objects. As a corollary, we obtain that there exists a Galois-equivariant isomorphism between their l-adic cohomology Frobenius algebras. We also discuss the case where the Kuznetsov component of a smooth cubic fourfold is Fourier-Mukai derived-equivalent to a K3 surface.
We show how the notion of the transcendence degree of a zero-cycle on a smooth projective variety X is related to the structure of the motive M(X). This can be of particular interest in the context of Blochs conjecture, especially for Godeaux surfaces, when the surface is given as a finite quotient of a suitable quintic in P^3.
We compute the Newton--Okounkov bodies of line bundles on the complete flag variety of GL_n for a geometric valuation coming from a flag of translated Schubert subvarieties. The Schubert subvarieties correspond to the terminal subwords in the decomposition (s_1)(s_2s_1)(s_3s_2s_1)(...)(s_{n-1}...s_1) of the longest element in the Weyl group. The resulting Newton--Okounkov bodies coincide with the Feigin--Fourier--Littelmann--Vinberg polytopes in type A.
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous conjectures concerning their dimensions, and generalizes the superset method.
We study vector bundles on flag varieties over an algebraically closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(2le dleq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where $k$ is an algebraically closed field of characteristic $p>0$. Let $E$ be a uniform vector bundle over $G$ of rank $rle d$. We show that $E$ is either a direct sum of line bundles or a twist of a pull back of the universal bundle $H_d$ or its dual $H_d^{vee}$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties $F(d_1,cdots,d_s)$ in characteristic zero are considered. We prove a structure theorem for bundles over flag varieties which are uniform with respect to the $i$-th component of the manifold of lines in $F(d_1,cdots,d_s)$. Furthermore, we generalize the Grauert-M$ddot{text{u}}$lich-Barth theorem to flag varieties. As a corollary, we show that any strongly uniform $i$-semistable $(1le ile n-1)$ bundle over the complete flag variety splits as a direct sum of special line bundles.
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