Related papers: Generalization of a criterion for semistable vecto…

Let $X$ be the Jacobian of a genus 2 curve $\widetilde{\mathcal{C}}$ over $\mathbb{C}$ and $Y$ be the associated Kummer surface. Consider an ample line bundle $L=O(m\widetilde{\mathcal{C}})$ on $X$ for an even number $m$, and its descent to…

We give an algebraic proof valid in arbitrary characteristic for the known equivalence between (strongly) slope semistable vector bundles with vanishing discriminant and vanishing determinant and numerically flat bundles. We also address a…

Let $S$ be an Enriques surface. In this paper we study the semistability of the restriction $\Omega_{S}|_C$ for a general curve $C \in |H|$, where $H$ is a globally generated and ample line bundle on $S$. We show that $\Omega_{S}|_C$ is…

Let $C$ be a chain-like curve having $n$ smooth components and $n-1$ nodes, where $n \geq 2$. Let $E$ be a vector bundle on $C$ and $V \subseteq H^0(E)$ be a linear subspace generating $E$. We investigate the (semi)stability of the kernel…

We generalize a construction of Hitchin to prove that, given any compact K\"ahler manifold $M$ with positive holomorphic sectional curvature and any holomorphic vector bundle $E$ over $M$, the projectivized vector bundle ${\mathbb P}(E)$…

In this article we study a special class of vector bundles, called tensors. A tensor consists of a vector bundle $E$ over a smooth irreducible projective variety and a morphism of vector bundles $\varphi$. As for classical vector bundles,…

We study when a smooth variety $X$, embedded diagonally in its Cartesian square, is the zero scheme of a section of a vector bundle of rank $\dim(X)$ on $X\times X$. We call this the diagonal property (D). It was known that it holds for all…

In this paper we consider the complex vector spaces of holomorphic cross-sections of homogeneous holomorphic vector bundles over elliptic adjoint orbits, and provide a sufficient condition for the vector spaces to be finite dimensional in…

Let $E$ be a uniform bundle on an arbitrary generalised Grassmannian $X$ defined over $\mathbb{C}$. We show that if the rank of $E$ is at most $e.d.(\mathrm{VMRT})$, then $E$ necessarily splits. For some generalised Grassmannians, we prove…

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…

We show that Horrock's criterion for the splitting of vector bundles on $\PP^n$ can be extended to vector bundles on multiprojective spaces and to smooth projective varieties with the weak CM property (see Definition 3.11). As a main tool…

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 this paper we show that on a general hypersurface of degree $r=3,4,5,6$ in ${\bf P}^5$ a rank 2 vector bundle $E$ splits if and only if $h^1 E(n)=h^2 E(n)=0$ for all $n \in \bf Z$.

A new construction of a universal connection was given in \cite{BHS}. The main aim here is to explain this construction. A theorem of Atiyah and Weil says that a holomorphic vector bundle $E$ over a compact Riemann surface admits a…

Criteria for a diffeomorphism of a smooth manifold $M$ to be lifted to a linear automorphism of a given real vector bundle $p\colon V\rightarrow M$, are stated. Examples are included and the metric and complex vector-bundle cases are also…

Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…

The construction for nonreduced projective moduli scheme of semistable admissible pairs is performed. We establish the relation of this moduli scheme with reduced moduli scheme built up in the previous article and prove that nonreduced…

Let $f:X\rightarrow Y$ be a generically smooth morphism between irreducible smooth projective curves over an algebraically closed field of arbitrary characteristic. We prove that the vector bundle $((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is…

We study the cohomological classification of vector bundles on smooth real affine surfaces and threefolds. We show that, as was observed in joint work in A. Asok and J. Fasel and in a coming joint paper with S. Banerjee and J. Fasel, under…

Fujita's second theorem for K\"ahler fibre spaces over a curve asserts that the direct image $V$ of the relative dualizing sheaf splits as the direct sum $ V = A \oplus Q$, where $A$ is ample and $Q$ is unitary flat. We focus on our…