Related papers: An elementary proof that vector bundles on $\mathb…

Distributions, i.e., subsets of tangent bundles formed by piecing together subspaces of tangent spaces, are commonly encountered in the theory and application of differential geometry. Indeed, the theory of distributions is a fundamental…

There is a long-standing conjecture which states that every uniform algebraic vector bundle of rank $r<2n$ on the $n$-dimensional projective space $\mathbb{P}^n$ over an algebraically closed field of characteristic $0$ is homogeneous. This…

Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…

Let $G$ be a reductive group acting on an affine scheme $V$. We study the set of principal $G$-bundles on a smooth projective curve $\mathcal C$ such that the associated $V$-bundle admits a section sending the generic point of $\mathcal C$…

We give a proof of Mukai's Theorem on the existence of certain exceptional vector bundles on prime Fano threefolds. To our knowledge this is the first complete proof in the literature. The result is essential for Mukai's biregular…

We investigate stratified-algebraic vector bundles on a real algebraic variety X. A stratification of X is a finite collection of pairwise disjoint, Zariski locally closed subvarieties whose union is X. A topological vector bundle on X is…

We construct an I-function of the projective bundle P(V) associated with a not necessarily split vector bundle V\to B as a Fourier transform of the S^1-equivariant J-function of the total space of V and show that it lies on the Givental…

Let A be a union of smooth plane curves C_i, such that each singular point of A is quasihomogeneous. We prove that if C is a smooth curve such that each singular point of A U C is also quasihomogeneous, then there is an elementary…

Given a curve in a (smooth) projective variety $C \subset X$, we show that a vector bundle $V$ on C can be extended to a ($\mu$-stable) vector bundle on $X$ if $\text{rank}(V) \geq \text{dim}(X)$ and $\text{det}(V)$ extends to $X$.

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…

We study the orientability of vector bundles with respect to a family of cohomology theories called $\mathrm{EO}$-theories. The $\mathrm{EO}$-theories are higher height analogues of real $\mathrm{K}$-theory $\mathrm{KO}$. For each…

A generalized Euler sequence over a complete normal variety X is the unique extension of the trivial bundle V \otimes O_X by the sheaf of differentials \Omega_X, given by the inclusion of a linear space V in Ext^1(O_X,\Omega_X). For…

The (parallel) linear transports along paths in vector bundles are axiomatically described. Their general form and certain properties are found. It is shown that these transports are locally (i.e. along every fixed path) always Euclidean…

We formulate a conjecture on the behavior of the minimal free resolutions of sets of general points on arbitrary varieties embedded by complete linear series, in analogy with the well-known Minimal Resolution Conjecture for points in…

Let $\pi\,:\, X \,\longrightarrow\, Y$ be a finite morphism of smooth projective varieties defined over an algebraically closed field of characteristic zero. We study the necessary and sufficient criteria for $\pi$ such that there exists a…

We show that on a generic curve and under some conditions on the degree and genus, there exists a component B of the locus of stable vector bundles of rank r and degree d with at least k sections of the expected dimension such that for a…

We discuss the projectivity of the moduli space of semistable vector bundles on a curve of genus $g\geq 2$. This is a classical result from the 1960s, obtained using geometric invariant theory. We outline a modern approach that combines the…

We show that any continuous $\mathbf{C}$-linear Lie algebra splitting of the symbol map from the Atiyah algebra of a vector bundle on a complex manifold is given by a differential operator of order at most the rank of the bundle plus one.…

In this paper we study the problem of extension of holomorphic sections of line bundles/vector bundles from reduced unions of strata of divisors. An extension theorem of Ohsawa--Takegoshi type is proved. As consequences we deduce several…

We generalise the variant of the Babylonian tower theorem for vector bundles on projective spaces proved by I. Coanda and G. Trautmann (2006) to the case of principal $G$-bundles over projective spaces, where $G$ is a linear algebraic group…