On Moduli of G-bundles over Curves for exceptional G

In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singular versions of them) over smooth projective varieties over algebraically closed ground fields of positive characteristic.

We show that principal bundles for a semisimple group on an arbitrary affine curve over an algebraically closed field are trivial, provided the order of $\pi_1$ of the group is invertible in the ground field, or if the curve has semi-normal singularities. Several consequences and extensions of this result (and method) are given. As an application, we realize conformal blocks bundles on moduli stacks of stable curves as push forwards of line bundles on (relative) moduli stacks...

For a simple complex Lie group G the connected components of the moduli space of G-bundles over an elliptic curve are weighted projective spaces. In this note we will provide a new proof of this result using the invariant theory of Kac-Moody groups, in particular the action of the (twisted) Coxeter element on the root system of G.

We prove the existence of a projective good moduli space of principal $\mathcal{G}$-bundles under nonconnected reductive group schemes $\mathcal{G}$ over a smooth projective curve $C$. We also prove that the moduli stack of $\mathcal{G}$-bundles decomposes into finitely many substacks $\text{Bun}_{\mathcal{P}}$ each admitting a torsor $\text{Bun}_{\mathcal{G}_\mathcal{P}}\to \text{Bun}_{\mathcal{P}}$ under a finite group, for some connected reductive group schemes $\mathcal{G...

It is well-known that del Pezzo surfaces of degree $9-n$ one-to-one correspond to flat $E_n$ bundles over an elliptic curve. In this paper, we construct $ADE$ bundles over a broader class of rational surfaces which we call $ADE$ surfaces, and extend the above correspondence to all flat $G$ bundles over an elliptic curve, where $G$ is any simply laced, simple, compact and simply-connected Lie group. In the sequel, we will construct $G$ bundles for non-simply laced Lie group $G...

Let $G$ be a semisimple complex algebraic group with a simple Lie algebra $\mathfrak{g}$, and let $\mathcal{M}^0_{G}$ denote the moduli stack of topologically trivial stable $G$-bundles on a smooth projective curve $C$. Fix a theta characteristic $\kappa$ on $C$ which is even in case $\dim{\mathfrak{g}}$ is odd. We show that there is a nonempty Zariski open substack ${\mathcal U}_\kappa$ of $\mathcal{M}^0_{G}$ such that $H^i(C,\, \text{ad}(E_G)\otimes\kappa) \,=\, 0$, $i\,=\,...

In this paper, which is a sequel of arXiv:2002.07494, we investigate, for any reductive group $G$ over an algebraically closed field $k$, the Picard group of the universal moduli stack $\mathrm{Bun}_{G,g,n}$ of $G$-bundles over $n$-pointed smooth projective curves of genus $g$. In particular: we give new functorial presentations of the Picard group of $\mathrm{Bun}_{G,g,n}$; we study the restriction homomorphism onto the Picard group of the moduli stack of principal $G$-bundl...

In this note, we introduce the notion of a singular principal G-bundle, associated to a reductive algebraic group G over the complex numbers by means of a faithful representation $\varrho^\p\colon G\lra \SL(V)$. This concept is meant to provide an analogon to the notion of a torsion free sheaf as a generalization of the notion of a vector bundle. We will construct moduli spaces for these singular principal bundles which compactify the moduli spaces of stable principal bundles...

Let $\mathcal C$ be a smooth irreducible projective curve over the complex numbers and let $G$ be a simple simply-connected complex algebraic group. Let $\mathfrak M=\mathfrak M(G,\mathcal C)$ be the moduli space of semistable principal $G$-bundles on $\mathcal C$. By an earlier result of Kumar-Narasimhan, the Picard group of $\mathfrak M$ is isomorphic with the group of integers. However, in their work the generator of the Picard group was not determined explicitly. The aim ...

Let H be a semisimple algebaric group and let X be a smooth projective curve defined over an algebraically closed field k. In the first part of this paper we show that the moduli of semistable principal H-bundles exists once given a "low-height" representation of H. We also show the projectivity of the moduli space if p > \psi, where \psi is a representation theoritic index. The projectivity is a consequence of a semistable reduction theorem. The irreducibility of the moduli ...