Using the Berline-Vergne integration formula for equivariant cohomology for torus actions, we prove that integrals over Grassmannians (classical, Lagrangian or orthogonal ones) of characteristic classes of the tautological bundle can be expressed as iterated residues at infinity of some holomorphic functions of several variables. The results obtained for these cases can be expressed as special cases of one formula involving the Weyl group action on the characters of the natural representation of...

Let $G$ be a split semisimple linear algebraic group over a field and let $T$ be a split maximal torus of $G$. Let $𝗁$ be an oriented cohomology (algebraic cobordism, connective $K$-theory, Chow groups, Grothendieck’s $K_0$, etc.) with formal group law $F$. We construct a ring from $F$ and the characters of $T$, that we call a formal group ring, and we define a characteristic ring morphism $c$ from this formal group ring to $𝗁(G/B)$ where $G/B$ is the variety of Borel subgroups of $G$. Our main result says that when the torsion index...

A fundamental result of Beĭlinson–Ginzburg–Soergel states that on flag varieties and related spaces, a certain modified version of the category of $\ell$-adic perverse sheaves exhibits a phenomenon known as Koszul duality. The modification essentially consists of discarding objects whose stalks carry a nonsemisimple action of Frobenius. In this paper, we prove that a number of common sheaf functors (various pull-backs and push-forwards) induce corresponding functors on the modified category or its triangulated...

Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of maximal multiplicity vanishing in Frobenius splitting. In this paper we define the algebraic analogue of this concept and construct a Frobenius splitting vanishing with maximal multiplicity on the diagonal of the full flag variety. Our splitting induces a diagonal Frobenius splitting of maximal multiplicity for a special class of smooth Schubert varieties first considered by Kempf. Consequences are Frobenius splitting of tangent bundles,...

A geodesic of a homogeneous Riemannian manifold $(M=G/K,g)$ is called homogeneous if it is an orbit of an one-parameter subgroup of $G$. In the case when $M=G/H$ is a naturally reductive space, that is the $G$-invariant metric $g$ is defined by some non degenerate biinvariant symmetric bilinear form $B$, all geodesics of $M$ are homogeneous. We consider the case when $M=G/K$ is a flag manifold, i.eȧn adjoint orbit of a compact semisimple Lie group $G$, and we give a simple necessary condition that $M$ admits a non-naturally reductive...

Let $F$ be a field and $Gr\left(i,F^n\right)$ be the Grassmannian of $i$-dimensional linear subspaces of $F^n$. A map $f:Gr\left(i,F^n\right)\to Gr\left(j,F^n\right)$ is called nesting if $l\subset f\left(l\right)$ for every $l\in Gr\left(i,F^n\right)$. Glover, Homer and Stong showed that there are no continuous nesting maps $Gr\left(i,{\mathbb{C}}^n\right)\to Gr\left(j,{\mathbb{C}}^n\right)$ except for a few obvious ones. We prove a similar result for algebraic nesting maps $Gr\left(i,F^n\right)\to Gr\left(j,F^n\right)$, where $F$ is an algebraically closed field of arbitrary characteristic. For $i=1$ this yields a description of the algebraic sub-bundles of the tangent bundle to the projective space $P_F^n$.

Inspired by Manin’s approach towards a geometric interpretation of Arakelov theory at infinity, we interpret in this paper non-Archimedean local intersection numbers of linear cycles in ${\mathbb{P}}^{n-1}$ with the combinatorial geometry of the Bruhat-Tits building associated to $PGL\left(n\right)$.