Related papers: Some Remarks on the Local Fundamental Group Scheme

We define locally trivial quantum vector bundles (QVB) and QVB associated to locally trivial quantum principal fibre bundles. There exists a differential structure on the associated vector bundle coming from the differential structure on…

We study the properness of the functor of $F$-trivial bundles by relating it to the base change question for the fundamental group scheme of Nori.

We provide for all prime numbers $p$ examples of smooth projective curves over a field of characteristic $p$ for which base change of the fundamental group scheme fails. This is intimately related to how $F$-trivial vector bundles, i.e.…

We define the algebraic fundamental group functor of a reductive group scheme over an arbitrary (non-empty) base scheme and prove that this functor is exact.

We prove that Local Fundamental Group Scheme satisfies the conditions of Lefschetz-Bott-Grothendieck.

We extend the definition of fundamental group scheme to non reduced schemes over any connected Dedekind scheme. Then we compare the fundamental group scheme of an affine scheme with that of its reduced part.

Let G be a simple simply-connected group scheme over a regular local scheme U. Let E be a principal G-bundle over A^1_U trivial away from a subscheme finite over U. We show that E is not necessarily trivial and give some criteria of…

We introduce the notion of locally trivial quantum principal bundles. The base space and total space are compact quantum spaces (unital $C^{\star}$-algebras), the structure group is a compact matrix quantum group. We prove that a quantum…

A categorical principal bundle is a structure comprised of categories that is analogous to a classical principal bundle; examples arise from geometric contexts involving bundles over path spaces. We show how a categorical principal bundle…

We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the (free) fundamental…

Let X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the…

For a weak 2-group, we construct a bicategory of flat 2-group bundles over differentiable stacks as a localization of a functor bicategory. This description is amenable to explicit geometric constructions. For example, we show that flat…

Let F be the function field of a curve over a complete discretely valued field K. Let G be a semisimple simply connected linear algebraic group over F of type An. We give a description of the obstruction to local global principle for…

We define the notion of holonomy group for a stable vector bundle F on a variety in terms of the Narasimhan--Seshadri unitary representation of its restriction to curves. Next we relate the holonomy group to the minimal structure group and…

The S-fundamental group scheme is the group scheme corresponding to the Tannaka category of numerically flat vector bundles. We use determinant line bundles to prove that the S-fundamental group of a product of two complete varieties is a…

It is shown that the principle of locality and noncommutative geometry can be connnected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. With the language of…

We prove that the Local Fundamental Group Scheme satisfies the Lefschetz - Bott theorems in characteristic p. The proofs are standard applications of the Enriques-Severi -Zariski-Serre vanishing theorems and known facts about the…

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…

Let $X$ be a projective and smooth variety over an algebraically closed field $k$. Let $f:Y\rightarrow X$ be a proper and surjective morphism of $k$-varieties. Assuming that $f$ is separable, we prove that the Tannakian category associated…

Let K be a p-adic field and F the function field of a curve over K. Let G be a connected linear algebraic group over F of classical type. Suppose the prime p is a good prime for G. Then we prove that projective homogeneous spaces under G…