Related papers: On endomorphisms of projective bundles

Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to \Sigma$ over a compact complex curve $\Sigma$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$…

We show that principal bundles for a semisimple group on an arbitrary affine curve over an algebraically closed field are trivial, provided the order of $\pi_1$ of the group is invertible in the ground field, or if the curve has semi-normal…

A vector bundle E on a projective variety X is called finite if it satisfies a nontrivial polynomial equation with integral coefficients. A theorem of Nori implies that E is finite if and only if the pullback of E to some finite etale…

We construct a tilting object for the stable category of vector bundles on a weighted projective line X of type (2,2,2,2;\lambda), consisting of five rank two bundles and one rank three bundle, whose endomorphism algebra is a canonical…

We show that a principal G bundle on a smooth projective curve over a finite field is strongly semistable if and only if it is defined by a representation of the fundamental group scheme of the curve into G.

We prove a number of results to the general effect that, under obviously necessary numerical and determinant constraints, "most" morphisms between fixed bundles on a complex elliptic curve produce (co)kernels which can either be specified…

Let $\Sigma$ be an orientbale closed surface and let $\Sigma'$ be a nonorientable closed surface. In the paper, we show that for any nontrivial orientable $S^2$ fiber bundles $X= \Sigma \ltimes S^2$ and $X' = \Sigma' \ltimes S^2$, there are…

This submission replaces the arXiv:1012.5381 submission with the same title, which had been withdrawn as it contained a mistake, repaired in this submission: on $X$ projective smooth over an algebraically closed field of characteristic…

Let $\alpha: X \to Y$ be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under $\alpha$ is semistable if the genus of $Y$ is at least $1$ and stable if the genus of $Y$ is at least $2$.…

We describe when two multiprojective bundles (fibre products of projective bundles over the same base) over projective spaces are isomorphic as abstract varieties. We also describe when two relative symmetric powers of projective bundles…

We prove by induction on dimension the Hodge conjecture for smooth complex projective varieties. Let $X$ be a smooth complex projective variety. Then $X$ is birational to a possibly singular projective hypersurface, hence to a smooth…

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 $d = 6$.…

Consider an absolutely simple abelian variety X over a number field K. If the absolute endomorphism ring of X is commutative and satisfies certain parity conditions, then the reduction X_p is absolutely simple for almost all p. Conversely,…

Let $X$ be a smooth Fano fourfold admitting a conic bundle structure. We show that $X$ is toric if and only if $X$ admits an amplified endomorphism; in this case, $X$ is a rational variety.

Given a covering f: X \to Y of projective manifolds, we consider the vector bundle E on Y given as the dual of f_*(\O_X) / \O_Y. This vector bundles often has positivity properties, e.g. E is ample when Y is projective space by a theorem of…

We prove that a generic Steiner bundle E is simple if and only if the Euler characteristic of the endomorphism bundle of E is less or equal to 1. In particular we show that either E is exceptional or it satisfies the following inequality…

Let $\pi: X \longrightarrow C$ be a fibration with reduced fibers over a curve $C$ and consider a polarization $H$ on the surface $X$. Let $E$ be a stable vector bundle of rank $2$ on $C$. We prove that the pullback $\pi^*E$ is a $H-$stable…

Smooth complex surfaces polarized with an ample and globally generated line bundle of degree three and four, such that the adjoint bundle is not globally generated, are considered. Scrolls of a vector bundle over a smooth curve are shown to…

We study simply-laced simple affine Lie algebra bundles over complex surfaces X. Given any Kodaira curve C in X, we construct such a bundle over X. After deformations, it becomes trivial on every irreducible component of C provided that…

For a smooth projective complex variety $X$, we prove that there exists a birational morphism $X\times X\to Y$ to a projective variety $Y$ contracting the diagonal $\Delta_X\subset X\times X$ to a point if and only if $X$ has maximal…