On the motive of certain subvarieties of fixed flags

We compute de Chow motive of certain subvarieties of the flags manifold and show that it is an Artin motive.

💡 Research Summary

The paper investigates the Chow motive of a class of subvarieties inside the full flag manifold that are defined by fixing a reference flag and imposing a collection of incidence conditions. Let ( \mathcal{F}\ell_n ) denote the complete flag variety over an algebraically closed field (k). Choose a fixed flag (F_{\bullet}) and two subsets (I,J\subset{1,\dots ,n}). The subvariety (X_{I,J}\subset\mathcal{F}\ell_n) consists of all flags (V_{\bullet}) satisfying the inclusions (F_i\subset V_j) for each pair ((i,j)) prescribed by (I) and (J). These are essentially intersections of Schubert varieties, and while they may be singular or non‑equidimensional, they are stable under a natural (\mathbb{G}_m)-action.

The authors first recall the Bruhat decomposition of (\mathcal{F}\ell_n) into Schubert cells, each isomorphic to an affine space. They then apply the Bialynicki‑Birula decomposition associated with the (\mathbb{G}m)-action to the subvariety (X{I,J}). The fixed points of the action are precisely those flags that simultaneously satisfy the incidence conditions; each fixed point gives rise to an attracting cell (C_{\lambda}) which is again an affine space (\mathbb{A}^{d_{\lambda}}). The dimensions (d_{\lambda}) are computed combinatorially from the data (I,J) using the length function on the Weyl group and the Bruhat order.

Having obtained an explicit cellular stratification, the paper passes to motives. Because each cell is an affine space, its Chow motive is the Tate motive (\mathbb{L}^{d_{\lambda}}) (where (\mathbb{L}) denotes the Lefschetz motive). Consequently the Chow motive of the whole subvariety decomposes as a direct sum of Tate motives: \