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We define vector bundles for tropical schemes, and explore their properties. The paper largely consists of three parts; (1) we study free modules over zero-sum free semirings, which provide the necessary algebraic background for the theory…

In an unpublished preprint, A. King conjectured that there are tilting bundles over projective varieties which are obtained as invariant quotients of affine spaces for linear actions of reductive groups. The goal of this paper is to give…

We study the representability of motivic spheres by smooth varieties. We show that certain explicit "split" quadric hypersurfaces have the $\mathbb A^1$-homotopy type of motivic spheres over the integers and that the $\mathbb A^1$-homotopy…

We study the notion of twisted bundles on noncommutative space. Due to the existence of projective operators in the algebra of functions on the noncommutative space, there are twisted bundles with non-constant dimension. The U(1) instanton…

We study the classification of affine holomorphic bundles over a compact complex manifold $X$ in general, and we apply the general theory to the case $X=\mathbb{P}^1_\mathbb{C}$. We study the moduli space of framed, non-degenerate rank 2…

Affine schemes can be understood as objects of the opposite of the category of commutative and unital algebras. Similarly, $\mathscr{P}$-affine schemes can be defined as objects of the opposite of the category of algebras over an operad…

We introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra $\mathfrak g$. These are sheaves on locally closed subvarieties of the…

In this paper we count the number of isomorphism classes of geometrically indecomposable quasi-parabolic structures of a given type on a given vector bundle on the projective line over a finite field. We give a conjectural cohomological…

In this note we highlight a common origin for many ubiquitous geometric structures, as well as several new ones by using only the functors of differential calculus in A.M Vinogradov's original sense, adapted to special classes of (graded)…

We construct stable vector bundles on the space of symmetric forms of degree d in n+1 variables which are equivariant for the action of SL_{n+1}(C), and admit an equivariant free resolution of length 2. For n=1, we obtain new examples of…

We study ample vector bundles on smooth projective stacks. In particular, we prove that the tangent bundle for the weighted projective stack $\mathbb{P}(a_0,...,a_n)$ is ample. A result of Mori shows that the only smooth projective…

Algebra bundles, in the strict sense, appear in many areas of geometry and physics. However, the structure of an algebra is flexible enough to vary non-trivially over a connected base, giving rise to a structure of a weak algebra bundle. We…

In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold $\mathbb{L}$, there exists a global trivialization of the tangent bundle, which defines a map…

Let $S$ be an irreducible smooth projective surface defined over an algebraically closed field $k$. For a positive integer $d$, let ${\rm Hilb}^d(S)$ be the Hilbert scheme parametrizing the zero-dimensional subschemes of $S$ of length $d$.…

In this paper we generalise the theory of real vector bundles to a certain class of non-Hausdorff manifolds. In particular, it is shown that every vector bundle fibred over these non-Hausdorff manifolds can be constructed as a colimit of…

In this paper, we define vector bundles within the framework of almost mathematics (referred to as almost vector bundles) and establish the $v$-descent theorem together with a structure theorem for these bundles over perfectoid spaces. The…

We obtain characterizations and structure results for homogeneous principal bundles over abelian varieties, that generalize work of Miyanishi and Mukai on homogeneous vector bundles. For this, we rely on notions and methods of algebraic…

We study moduli spaces of vector bundles on a two-dimensional neighbourhood $Z_k$ of an irreducible curve $\ell = CP^1$ with $\ell^2 = -k$ and give an explicit construction of these moduli as stratified spaces. We give sharp bounds for the…

We describe derived moduli functors for a range of problems involving schemes and quasi-coherent sheaves, and give cohomological conditions for them to be representable by derived geometric n-stacks. Examples of problems represented by…

We describe deformations of vector bundles on surfaces that are a product of two smooth projective curves. We explicitly describe the Kuranishi map around unstable vector bundles and compare the homologies of the Kuranishi spaces of stable…