Related papers: Constructing projective varieties in weighted flag…

We study algebraic geometry linear codes defined by linear sections of the Grassmannian variety as codes associated to FFN$(1,q)$-projective varieties. As a consequence, we show that Schubert, Lagrangian-Grassmannian, and isotropic…

We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical…

Using geometrical correspondences induced by projections and two-steps flag varieties, and a generalization of Orlov's projective bundle theorem, we relate the Hodge structures and derived categories of subvarieties of different…

In this paper, we solve the initial value problems of variable-coefficient generalized wave equations associated with trees and a large family of linear constant-coefficient partial differential equation by algebraic methods. Moreover, we…

In this note, we give Gysin formulas for partial flag bundles for the classical groups. We then give Gysin formulas for Schubert varieties in Grassmann bundles, including isotropic ones. All these formulas are proved in a rather uniform way…

We construct Koppelman formulas on manifolds of flags in $\C^N$ for forms with values in any holomorphic line bundle as well as in the tautological vector bundles and their duals. As an application we obtain new explicit proofs of some…

Let G be a complex semi-simple Lie group and let P,Q be a pair of parabolic subgroups of G such that Q contains P. Consider the flag varieties G/P, G/Q and Q/P. We show that certain structure constants in H^*(G/P) with respect to the…

We relate properties of weighted flags (or multiflags) of type AD to statistics of the corresponding Weyl groups. For type A, we recover the Mahonian statistics on symmetric groups. Finally, we sketch briefly an easy extension incorporating…

We give two simple geometric constructions of a smooth family of projective varieties with central fiber isomorphic to the horospherical variety of type $\mathrm{G}_2$ and all other fibers isomorphic to the isotropic orthogonal Grassmannian…

In this paper we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand-Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle…

We present a review on the recently discovered link between the Lie theory,the theory of quiver Grassmannians, and various degenerations of flag varieties. Our starting point is the induced Poincar\'{e}--Birkhoff--Witt filtration on the…

Let $\mathbb{F}$ be a field of characteristic zero and let $\mathfrak{g}$ be a non-zero finite-dimensional split semisimple Lie algebra with root system $\Delta$. Let $\Gamma$ be a finite set of integral weights of $\mathfrak{g}$ containing…

We use incidence relations running in two directions in order to construct a Kempf-Laksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian.…

We define the odd symplectic grassmannians and flag manifolds, which are smooth projective varieties equipped with an action of the odd symplectic group and generalizing the usual symplectic grassmannians and flag manifolds. Contrary to the…

We introduce symplectic structures on "Lie pairs" of (real or complex) algebroids as studied by Chen, Stienon and the second author (From Atiyah classes to homotopy Leibniz algebras, arXiv:1204.1075), encompassing homogeneous symplectic…

We classify complex projective varieties of dimension $2r \geq 8$ swept out by a family of codimension two grassmannians of lines $\mathbb{G}(1,r)$. They are either fibrations onto normal surfaces such that the general fibers are isomorphic…

We introduce a superpotential for partial flag varieties of type $A$. This is a map $W: Y^\circ \to \mathbb{C}$, where $Y^\circ$ is the complement of an anticanonical divisor on a product of Grassmannians. The map $W$ is expressed in terms…

We compute the topological Witt groups of every complex flag manifold of ordinary type, and thus the interesting (i.e. torsion) part of the KO-groups of these manifolds. Equivalently, we compute Balmer's Witt groups of each flag variety of…

Lie theory is, beyond any doubt, an absolutely essential part of differential geometry. It is therefore necessary to seek its generalization to $\mathbb{Z}$-graded geometry. In particular, it is vital to construct non-trivial and explicit…

A special linear Grassmann variety SGr(k,n) is the complement to the zero section of the determinant of the tautological vector bundle over Gr(k,n). For a representable ring cohomology theory A(-) with a special linear orientation and…