Related papers: On connections on principal bundles

Consider an anchored bundle $(E,\rho)$, i.e. a vector bundle $E\to M$ equipped with a bundle map $\rho \colon E \to TM$ covering the identity. M.~Kapranov showed in the context of Lie-Rinehard algebras that there exists an extension of this…

This paper works as an appendix of the paper titled Geometry of Associated Quantum Vector Bundles and the Quantum Gauge Group and for paper titled Yang-Mills-Connes Theory and Quantum Principal SU(N)-Bundles. Here, we are going to prove…

We define functorial isomorphisms of parallel transport along etale paths for a class of G-principal bundles on a p-adic curve where G is a connected reductive algebraic group of finite presentation. This class consists of all principal…

We study infinitesimal deformations of autodual and hyper-holomorphic connections on complex vector bundles on hyper-K\"ahler manifolds of arbitrary dimension. In particular, we describe the DG Lie algebra controlling this deformation…

On an arbitrary compact Riemann surface, necessary and sufficient conditions are found for the existence of semistable vector bundles with slope between zero and one and a prescribed number of linearly independent holomorphic sections.…

We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) C*-algebras.…

Tangent categories provide an axiomatic approach to key structural aspects of differential geometry that exist not only in the classical category of smooth manifolds but also in algebraic geometry, homological algebra, computer science, and…

By virtue of the well-known theorem, a structure Lie group G of a principal bundle P is reducible to its closed subgroup H iff there exists a global section of the quotient bundle P/H. In gauge theory, such sections are treated as classical…

We formalize the ``metric bundle'' viewpoint by defining, for any smooth $n$--manifold $M$, the open fiberwise cones $\mathcal{G}^{p,q}\subset S^2\Tstar M$ of nondegenerate symmetric bilinear forms with fixed signature $(p,q)$, and we…

A vector bundle on a projective variety has a natural cohomology if for every twist its cohomology is concentrated in a single degree. Eisenbud and Schreyer conjectured there should be vector bundles on $\mathbb{P}^1 \times \mathbb{P}^1$…

We prove that a system of equations introduced by Demailly (to attack a conjecture of Griffiths) has a smooth solution for a direct sum of ample line bundles on a Riemann surface. We also reduce the problem for general vector bundles to an…

In this article we consider the continuity of the eigenvalues of the connection Laplacian of $G$-connections on vector bundles over Riemannian manifolds. To show it, we introduce the notion of the asymptotically $G$-equivariant measured…

Let $(M,g)$ be a smooth Anosov Riemannian manifold and $\mathcal{C}^\sharp$ the set of its primitive closed geodesics. Given a Hermitian vector bundle $\mathcal{E}$ equipped with a unitary connection $\nabla^{\mathcal{E}}$, we define…

A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold $X$ makes $T_X[-1]$ into a Lie algebra object in $D^+(X)$, the bounded below derived category of coherent sheaves on $X$. Furthermore…

In the present note we describe geometrically the homology classes in the total space of a surface bundle over a surface in terms of the holonomy map. We treat the cases where the base surface is closed or has one boundary component. We…

For the first obstruction to extending a holomorphic vector bundle from a submanifold of a complex manifold found by Griffiths we give an explicit formula in terms of the Atiyah class of the bundle.

Let M be an irreducible smooth projective variety defined over \bar{{\mathbb F}_p}. Let \pi(M, x_0) be the fundamental group scheme of M with respect to a base point x_0. Let G be a connected semisimple linear algebraic group over…

We define functorial isomorphisms of parallel transport along etale paths for a class of vector bundles on a p-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise…

Fujita's second theorem for K\"ahler fibre spaces over a curve asserts that the direct image $V$ of the relative dualizing sheaf splits as the direct sum $ V = A \oplus Q$, where $A$ is ample and $Q$ is unitary flat. We focus on our…

We define homology groups for flat irregular singular connections on surfaces and a pairing between these and the de Rham cohomology of the connection, generalizing work of S. Bloch and H. Enault in dimension one. Assuming a conjecture of…