Related papers: Locally trivial quantum vector bundles and associa…
We study the relations between the triviality of the tangent bundle $TM$ and the generalized tangent bundle $\mathbb{T}M = TM\oplus T^*M$ of a manifold. We show that the generalized tangent bundle of a paralellizable manifold is trivial. We…
We prove two results about vector bundles on singular algebraic surfaces. First, on proper surfaces there are vector bundles of rank two with arbitrarily large second Chern number and fixed determinant. Second, on separated normal surfaces…
We generalize the notion of parallel transport along paths for abelian bundles to parallel transport along surfaces for abelian gerbes using an embedded Topological Quantum Field Theory (TQFT) approach. We show both for bundles and gerbes…
We present equivalences between certain categories of vector bundles on projective varieties, namely cokernel bundles, Steiner bundles, syzygy bundles, and monads, and full subcategories of representations of certain quivers. As an…
For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the…
A general noncommutative-geometric theory of principal bundles is presented. Quantum groups play the role of structure groups. General quantum spaces play the role of base manifolds. A differential calculus on quantum principal bundles is…
A $\mathbb Q$-conic bundle is a proper morphism from a threefold with only terminal singularities to a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We study the structure of $\mathbb…
A noncommutative-geometric generalization of the theory of principal bundles is sketched. A differential calculus over corresponding quantum principal bundles is analysed. The formalism of connections is presented. In particular, operators…
Quantum principal bundles or principal comodule algebras are re-interpreted as principal bundles within a framework of Synthetic Noncommutative Differential Geometry. More specifically, the notion of a noncommutative principal bundle within…
In this paper we introduce the Integration Problem for principal connections. Just as a principal connection on a principal bundle $\phi:Q\rightarrow M$ may be used to split $TQ$ into horizontal and vertical subbundles, a discrete…
We study VB-groupoids and VB-algebroids, which are vector bundles in the realm of Lie groupoids and Lie algebroids. Through a suitable reformulation of their definitions, we elucidate the Lie theory relating these objects, i.e., their…
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…
We construct equivariant vector bundles over quantum projective spaces making use of parabolic Verma modules over the quantum general linear group. Using an alternative realization of the quantized coordinate ring of projective space as a…
We describe a locally trivial quantum principal U(1)-bundle over the quantum space S^2_{pq} which is a noncommutative analogue of the usual Hopf bundle. We also provide results concerning the structure of its total space algebra…
In other to study connections and gauge theories on noncommutative spaces it is useful to use the local trivializations of principal bundles. In this note we show how to use noncommutative localization theory to describe a simple version of…
A diffeological connection on a diffeological vector pseudo-bundle is defined just the usual one on a smooth vector bundle; this is possible to do, because there is a standard diffeological counterpart of the cotangent bundle. On the other…
We develop an intersection theory for a singular hemitian line bundle with positive curvature current on a smooth projective variety and irreducible curves on the variety. And we prove the existence of a natural rational fibration structure…
The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The…
We develop a theory of smooth relative connections over the real path algebra $\mathbb{R}Q$ on smooth twisted quiver bundles. We give obstructions to the existence of a smooth relative connection on twisted quiver bundles. For tree-type…
In this short article we review how the classical theory of principal fibre bundles (PFB) transcribes in an algebraic formalism. In this dual formulation, a PFB is given by a right co-module algebra ${\cal P}$ over a Hopf algebra ${\cal H}$…