Related papers: Toric principal bundles, piecewise linear maps and…

Let $(G,{\mathfrak I},N,S)$ be an affine topological Tits system, and let $\Gamma$ be a torsion free cocompact lattice in $G$. This article studies the coinvariants $H_0(\Gamma; C(\Omega,{\mathbb Z}))$, where $\Omega$ is the Furstenberg…

Let G be an affine reductive algebraic group over an algebraically closed field k. We determine the Picard group of the moduli stacks of principal G-bundles on any smooth projective curve over k.

Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical…

We develop a "Soergel theory" for Bruhat-constructible perverse sheaves on the flag variety $G/B$ of a complex reductive group $G$, with coefficients in an arbitrary field $\Bbbk$. Namely, we describe the endomorphisms of the projective…

Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x_0. Let \varpi(M,x_0) denote the corresponding fundamental group--scheme introduced by Nori. Let E_G be a…

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic zero carrying the trivial valuation. In this article we discuss two candidates for what could be the tropicalization of $G$. Our first…

Let X be a T-variety, where T is an algebraic torus. We describe a fully faithful functor from the category of T-equivariant vector bundles on X to a certain category of filtered vector bundles on a suitable quotient of X by T. We show that…

Let ${\mathcal B}_g(r)$ be the moduli space of triples of the form $(X,\, K^{1/2}_X,\, F)$, where $X$ is a compact connected Riemann surface of genus $g$, with $g\, \geq\, 2$, $K^{1/2}_X$ is a theta characteristic on $X$, and $F$ is a…

Given a vector bundle $A\to M$ we study the geometry of the graded manifolds $T^*[k]A[1]$, including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical…

Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B_1, ..., B_k on X and integers m_1, ..., m_k, consider the line bundle L :=…

For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex…

We consider the moduli space of vector bundles of rank $n$ and degree $ng$ over a fixed Riemann surface of genus $g\geq 2$. We use the explicit parametrization in terms of the Tyurin data. In the moduli space there is a "non-abelian" Theta…

If A is a strongly noetherian graded algebra generated in degree one, then there is a canonically constructed graded ring homomorphism from A to a twisted homogeneous coordinate ring B(X, L, sigma), which is surjective in large degree. This…

We give a geometric interpretation of the Weil representation of the metaplectic group, placing it in the framework of the geometric Langlands program. For a smooth projective curve X we introduce an algebraic stack \tilde\Bun_G of…

We exhibit full exceptional collections of vector bundles on any smooth, Fano arithmetic toric variety whose split fan is centrally symmetric.

The purpose of this paper is to extend the Donaldson-Corlette theorem to the case of vector bundles over cell complexes. We define the notion of a vector bundle and a Higgs bundle over a complex, and describe the associated Betti, de Rham…

For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral piecewise polynomial functions on P. We use the…

We outline a proof that the stack of symplectic toric G-manifolds over a fixed orbit space W is a torsor for the stack of symplectic toric G-principal bundles over W.

Given any topological group $G$, the topological classification of principal $G$-bundles over a finite CW-complex $X$ is long-known to be given by the set of free homotopy classes of maps from $X$ to the corresponding classifying space…

Using the notion of a root datum of a reductive group $G$ we propose a tropical analogue of a principal $G$-bundle on a metric graph. We focus on the case $G=\mathrm{GL}_n$, i.e. the case of vector bundles. Here we give a characterization…