Related papers: Cotangent Bundle to the Flag Variety - I
Let $\rm{Bun}$ be the moduli stack of rank $2$ bundles with fixed determinant on a smooth proper curve $C$ over a local field $F$. We show how to associate with a Schwartz $\kappa$-density, for $\rm{Re}(\kappa)\ge 1/2$, a smooth function on…
Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and…
We show that the totally nonnegative part of the twisted product of flag varieties of a Kac-Moody group admits a cellular decomposition, and the closure of each cell is a topological manifold with boundary. We also establish explicit…
The stratified vector bundles on a smooth variety defined over an algebraically closed field $k$ form a neutral Tannakian category over $k$. We investigate the affine group--scheme corresponding to this neutral Tannakian category.
We determine the structure of the equivariant cohomology and $K$-theory of Bott towers. By restriction, we obtain similar results for Bott-Samelson varieties. This results allow us to describe more precisely the equivariant cohomology and…
A gauge theoretic description of the Morgan-Shalen compactification of the $\SL$ character variety of the fundamental group of a hyperbolic surface is given in terms of a natural compactification of the moduli space of Higgs bundles via the…
In this paper we prove a formula relating the equivariant Euler characteristic of $K$-theoretic stable envelopes to an object known as the index vertex for the cotangent bundle of the full flag variety. Our formula demonstrates that the…
We study smooth projective complex varieties with ample cotangent bundle. Our main result is that in an abelian variety of dimension n, a complete intersection of at least n/2 general hypersurfaces of sufficiently high degrees has ample…
We study the $S_n$-equivariant log-concavity of the cohomology of flag varieties, also known as the coinvariant ring of $S_n$. Using the theory of representation stability, we give computer-assisted proofs of the equivariant log-concavity…
In this continuation of \cite{BM}, we prove the following: Let $\Gamma\subset \text{SL}(2,{\mathbb C})$ be a cocompact lattice, and let $\rho: \Gamma \rightarrow \text{GL}(r,{\mathbb C})$ be an irreducible representation. Then the…
We describe how each finite dimensional Schubert cell in the affine flag variety of $\text{SL}_2$ decomposes into orbits for a chain of subgroups of codimension one to four of the Iwahori group.
We compute the wonderful compactification of symmetric varieties of $SL(2,F)$, where $F$ is a finite field-extension of $Q_p$ with $p\neq 2$, that comes from either an abstract or $F$-involutions of $SL(2,F)$. For each of those wonderful…
We prove that the prime torsion in the local integral intersection cohomology of Schubert varieties in the flag variety of the general linear group grows exponentially in the rank. The idea of the proof is to find a highly singular point in…
Forgetting a subspace from a partial flag yields another partial flag composed of fewer subspaces. This induces a forgetful map $\pi : X \to X'$ between the corresponding flag varieties. We prove here that, for a degree large enough, the…
We study the hyperbolicity of compactifications of quotients of bounded symmetric domains by arithmetic groups. We prove that, up to an \'etale cover, they are Kobayashi hyperbolic modulo the boundary. Applying our techniques to Siegel…
For a semisimple Lie group $G_\mathbb{C}$ over $\mathbb{C}$, we study the homotopy type of the symplectomorphism group of the cotangent bundle of the flag variety and its relation to the braid group. We prove a homotopy equivalence between…
We give a "lattice-theoretic" description of the global Schubert variety for $\mathrm{GL}_n$ associated to any dominant coweight.
We construct a generalised notion of Kac-Moody algebras using smooth maps from the non-compact manifolds ${\cal M}=$SL$(2,\mathbb R)$ and ${\cal M}=$ SL$(2,\mathbb R)/U(1)$ to a finite-dimensional simple Lie group $G$. This construction is…
We study vector bundles on flag varieties over an algebraically closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(2\le d\leq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where…
This paper generalizes the results of the paper \cite{mi3} to the case of the general $\mathfrak{sl}_2$ Schubert varieties. We study the homomorphisms between different Schubert varieties, describe their geometry and the group of the line…