Related papers: Affine Gelfand-Dickey brackets and holomorphic vec…

We develop the theory of d-holomorphic connections on d-holomorphic vector bundles over a Klein surface by constructing the analogous Atiyah exact sequence for d-holomorphic bundles. We also give a criterion for the existence of…

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…

We prove that any holomorphic vector bundle admitting a holomorphic connection, over a compact K\"ahler Calabi-Yau manifold, also admits a flat holomorphic connection. This addresses a particular case of a question asked by Atiyah and…

Let $(V, \phi)$ be a holomorphic Lie algebroid over an irreducible smooth complex projective variety $X$ of dimension at least three, and let $E$ be a holomorphic vector bundle on $X$. We establish a necessary and sufficient condition for…

It is in general unknown which topological complex vector bundles on a non-algebraic surface admit holomorphic structures. We solve this problem for primary Kodaira surfaces by using results of Kani on curves of genus two with elliptic…

Given a projective morphism of compact, complex, algebraic varieties and a relatively ample line bundle on the domain we prove that a suitable choice, dictated by the line bundle, of the decomposition isomorphism of the Decomposition…

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 study vector bundles on the moduli stack of elliptic curves over a local ring R. If R is a field or a discrete valuation ring of (residue) characteristic not 2 or 3, all these vector bundles are sums of line bundles. For R the 3-local…

For $k \in \mathbb{Z}_{>0}$, let $\mathcal{H}^{(k)}_{g,n}$ denote the vector bundle over $\mathfrak{M}_{g,n}$ whose every fiber consists of meromorphic $k$-differentials with poles of order at most $k-1$ on a fixed Riemman surface of genus…

This paper gives various methods for constructing vector bundles over elliptic curves and more generally over families of elliptic curves. We construct universal families over generalized elliptic curves via spectral cover methods and also…

We study vector bundles with some additional structures on an elliptic curve and show how there are related to the elliptic Ruijsenaars-Schneider model.

We prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi-Yau threefold. For this purpose we prove the existence of certain stable vector bundle…

A tangent category is a category equipped with an endofunctor that satisfies certain axioms which capture the abstract properties of the tangent bundle functor from classical differential geometry. Cockett and Cruttwell introduced…

We prove the formality theorem for the differential graded Lie algebra module of Hochschild chains for the algebra of endomorphisms of a smooth vector bundle. We discuss a possible application of this result to a version of the algebraic…

We prove that stably isomorphic vector bundles of rank d-1 on a smooth affine d-fold X over an algebraically closed field k are indeed isomorphic, provided d! is invertible in k. This answers an old conjecture of Suslin.

A VB-algebroid is a vector bundle object in the category of Lie algebroids. We attach to every VB-algebroid a differential graded Lie algebra and we show that it controls deformations of the VB-algebroid structure. Several examples and…

We prove that the category of ``vector bundles on the absolute Fargues--Fontaine curve'' (more precisely the category of sections over some discrete algebraically closed field of the $v$-stack $\mathrm{Bun}_\mathrm{FF}$ of vector bundles on…

Let $X$ be a normal, connected and projective variety over an algebraically closed field $k$. It is known that a vector bundle $V$ on $X$ is essentially finite if and only if it is trivialized by a proper surjective morphism $f:Y\to X$. In…

Consider moduli schemes of vector bundles over a smooth projective curve endowed with parabolic structures over a marked point. Boden and Hu observed that a slight variation of the weights leads to a desingularisation of the moduli scheme,…

We completely classify all quotient bundles of a given vector bundle on the Fargues-Fontaine curve. As consequences, we have two additional classification results: a complete classification of all vector bundles that are generated by a…