arXiv research

This paper shows vector bundles and differential bundles are equivalent in smooth manifolds.

Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.

New pushforward operation on vector pseudo-bundles creates new examples.

Study the structure of linear bundle morphisms between vector bundles.

Theory of 2-vector bundles for smooth manifolds developed.

Article studies symmetry in smooth vector bundles using advanced operations.

Study on topological rigidity of ALE vector bundles with specific conditions.

We consider one possible definition of a diffeological connection on a diffeological vector pseudo-bundle. It is different from the one proposed in [7] and is in fact simpler, since it is obtained by a straightforward adaption of the standard definition of a connection as an operator on the space of all smooth sections…

Classifies equivariant vector bundles over toric manifolds.

Smooth actions of the multiplicative monoid $(\mathbb{R},\cdot)$ of real numbers on manifolds lead to an alternative, and for some reasons simpler, definition of a vector bundle, a double vector bundle and related structures like a graded bundle [Grabowski and Rotkiewicz, J. Geom. Phys. 2011]. For these reasons it is n…

Paper introduces stratified vector bundles and their properties.

Criteria for lifting manifold diffeomorphisms to vector bundle automorphisms.

A bundle gerbe is constructed from an oriented smooth vector bundle of even rank with a fiberwise inner product, over a compact connected orientable smooth manifold with Riemannian metric. From a trivialization of the bundle gerbe is constructed an irreducible Clifford module bundle, a spinor bundle over the smooth fre…

We prove a GAGA-style result for toric vector bundles with smooth base and give an algebraic construction of the Frölicher approximating vector bundle that has recently been introduced by Dan Popovici using analytic techniques.

This paper introduces $\infty$- and $n$-fold vector bundles as special functors from the $\infty$- and $n$-cube categories to the category of smooth manifolds. We study the cores and "n-pullbacks" of $n$-fold vector bundles and we prove that any $n$-fold vector bundle admits a non-canonical isomorphism to a decomposed …

New characterization of vector bundles using Lie algebras of symbols.

A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show …

The study examines connections and their curvatures on different types of bundles.

We give a proof of the existence of radial (smooth) parallel sections of vector bundles endowed with a linear connection.

New stability criteria for vector bundles linked to Hermite-Einstein geometry.

Study perturbations of submodules in Drury-Arveson space, finding smooth vector bundles with Hermitian connections.

Real vector bundles are determined by their Dirac indices on specific spin manifolds.

We develop the theory of smooth principal bundles for a smooth group $G$, using the framework of diffeological spaces. After giving new examples showing why arbitrary principal bundles cannot be classified, we define $D$-numerable bundles, the smooth analogs of numerable bundles from topology, and prove that pulling ba…

The paper proves stability of pulled back parabolic bundles on curves.

We consider when a smooth vector bundle endowed with a connection possesses non-trivial, local parallel sections. This is accomplished by means of a derived flag of subsets of the bundle. The procedure is algebraic and rests upon the Frobenius Theorem.

The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$ consists of all equivalent classes of curves that agree up to their accelerations of order $k$. For a Banach manifold $M$ and a natural number $k$ first we determine a smooth manifold structure on $T^kM$ which also offers a fiber bundle structure f…

The paper explores smooth equivariant rigidity and finds infinitely many exotic smooth structures.

The Hodge Conjecture is equivalent to a statement about conditions under which a complex vector bundle on a smooth complex projective variety admits a holomorphic structure. I advertise a class of abelian four-folds due to Mumford where this approach could be tested. I construct explicit smooth vector bundles - which c…

Generalizes Higgs bundles theory using a vector bundle twist.

In this paper, we show that if the tangent bundle of a smooth projective variety is strictly nef, then it is isomorphic to a projective space; if a projective variety $X^n$ $(n>4)$ has strictly nef $Λ^2 TX$, then it is isomorphic to $\mathbb{P}^n$ or quadric $\mathbb{Q}^n$. We also prove that on elliptic curves, strict…

Computes derivatives of sections in vector bundles using Lie derivatives.

We prove that smooth 1-dimensional topological field theories over a manifold are equivalent to vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rather than gluing laws. We make this idea precise through a smooth version of Rezk'…

Let M be a paracompact smooth manifold, A a Weil algebra and M^A the associated Weil bundle. In this paper, we give another definition and characterization of vector field on M^A.

Extends Serre-Swan theorem to all finitely generated modules over smooth functions.

We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang-Mills theory over $S ^{2}$ to show that any non-trivial, smooth Hermitian vector bundle $E$ over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-tri…

Extends Gelfand duality to various geometric and analytical categories.

Study shows how Poisson brackets factor on infinite dimensional manifolds.

Proofs for flows of linear vector fields and their applications.

We prove that all smooth sphere bundles that admit fiberwise 1/4-pinched metrics are induced bundles of vector bundles, so their structure groups reduce from the diffeomorphism group of the sphere to the orthogonal group. This result implies the existence of many smooth n-sphere bundles over a k-sphere that do not supp…

A uniqueness result in the inverse problem for an inhomogeneous hyperbolic system on a real vector bundle over a smooth compact manifold, based on energy measurements for improperly known sources, is established.

The paper studies metrics on vector bundles with singularities and their associated forms.

First, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

For each holomorphic vector bundle we construct a holomorphic bundle 2-gerbe that geometrically represents its second Beilinson-Chern class. Applied to the cotangent bundle, this may be regarded as a higher analogue of the canonical line bundle in complex geometry. Moreover, we exhibit the precise relationship between …

Researchers solve the Calderón problem for fractional Dirac operators.

Let $X$ be a smooth projective variety over $\mathbb C$. We prove that a twisted Higgs vector bundle $(\calE\, ,θ)$ on $X$ admits an Einstein--Hermitian connection if and only if $(\calE\, ,θ)$ is polystable. A similar result for twisted vector bundles (no Higgs fields) was proved by S. Wang in \cite{Wa}. Our approach …

We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the…

The paper defines a Chern-Simons invariant for stably trivial vector bundles and uses it to obstruct conformal immersions.

The paper characterizes vector bundles and differential operators using Lie algebras and their symbols.