Related papers: Extension de fibres vectoriels et profondeur

Grothendieck-Birkhoff Theorem states that every finite dimensional vector bundle over the projective line P1 splits as the sum of one dimensional vector bundles. This can be rephrased, in terms of orders, as stating that all maximal orders…

We give a generalization, in the context of sheaves, of a classical result of Grothendieck concerning the integrability of connections of type $(0,1)$ over a ${\cal C}^{\infty}$ vector bundle over a complex manifold. We introduce the notion…

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

Let X be a projective smooth irreducible polarized variety over the field of complex numbers. Typical examples of wide extensions are vector bundles E that have a subsheaf F whose slope is much bigger than the slope of E/F, and such that F…

This note contains another proof of Grothendieck`s theorem on the splitting of vector bundles on the projective line over a field $k$. Actually the proof is formulated entirely in the classical terms of a lattice $\Lambda \cong k[T]^d$,…

The paper explores some algebraic constructions arising in the theory of Lefschetz fibrations. Specifically, it covers in a fair amount of detail the algebraic issues outlined in ``Symplectic homology as Hochschild homology''…

Since Schwarzenberger and his celebrated paper called "Vector bundles on the projective plane" we know that any rank two vector bundle on $\P^2$ is a direct image of a line bundle on a double covering of the plane. This theorem suggests to…

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of…

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…

At the end of the twentieth century, J.-B. Bost developped a slope theory of hermitian vector bundles over number fields. A new method of diophantine approximation, the so-called slope method, has emerged from his research. Our article…

The question of existence of Ulrich bundles on nonsingular projective varieties is posed here in weaker terms: either to find a K-theoretic solution, or to find one in the derived category of the variety. We observe that if any motivic…

Grothendieck-Birkhoff Theorem states that every finite dimensional vector bundle over the projective line splits as the sum of one dimensional vector bundles. In this work we study simultaneous splittings of two dimensional vector bundles…

These notes aim at providing a complete and systematic account of some foundational aspects of algebraic supergeometry, namely, the extension to the geometry of superschemes of many classical notions, techniques and results that make up the…

We address the question of existence of sections of fibrations in two settings. First, we show that a bundle with base a finite 2-complex admits a section if and only if the inclusion of the fiber is $\pi_1$-injective and the associated…

We study in this paper a new family of stable algebraic symplectic vector bundles of rank $ 2n $ on the complex projective space $P^{2n+1}$ whose classical null correlation bundles belongs. We show that these bundles are invariant under a…

In this paper, we investigate linear systems on hyperelliptic varieties. We prove analogues of well-known theorems on abelian varieties, like Lefschetz's embedding theorem and higher k-jet embedding theorems. Syzygy or $N_p$-properties are…

We prove a Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles $E$ and $F$ over a complex manifold under the condition that $E^*\ox F$ is Griffiths $k$-positive. We apply this result…

For a field of characteristic $\ne 2$ we study vector spaces that are graded by the weight lattice of a root system, and are endowed with linear operators in each simple root direction. We show that these data extend to a graded semisimple…

We investigate an interplay between some ideas in traditional gauge theory and certain concepts in fibered categories. We accomplish this by introducing a notion of a principal Lie 2-group bundle over a Lie groupoid and studying its…

A degree $d$ genus $g$ cover of the complex projective line by a smooth irreducible curve $C$ yields a vector bundle on the projective line by pushforward of the structure sheaf. We classify the bundles that arise this way when $d = 6$.…