Let X be a nonsingular algebraic curve of genus g ≥ 3, and M ξ the moduli space of stable vector bundles of rank n ≥ 2 and degree d with fixed determinant ξ over X such that n and d coprime and d > n(2g − 2). We assume that if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3. Let W ξ (L) denote the vector bundle over M ξ defined as the direct image π * (U ξ ⊗ p * 1 L) where U ξ is a universal vector bu...

We develop an alternative view on the concept of connections over a vector bundle map, which consists of a horizontal lift procedure to a prolonged bundle. We further focus on prolongations to an affine bundle and introduce the concept of affineness of a generalised connection.

We determine in an elementary fashion the dimension of the space of innni-tesimal deformations of an arbitrary vector bundle on the complex projective line, showing its dependence on the precise form of the decomposition of the bundle under consideration.

Let £ be a vector bundle and P(E) its associated projective bundle. Some necessary conditions on the characteristic classes of E for existence of a fibrewise fixed-point free map P(E) -> P(E) are obtained.

In this paper we give the classification of rank 3 vector bundles without ”inner” cohomology on a quadric hypersurface Qn (n > 3) by studying the associated monads. Introduction A monad on P or, more generally, on a projective variety X, is a complex of three vector bundles 0 → A α −→ B β −→ C → 0 such that α is injective as a map of vector bundles and β is surjective. Monads have been studied ...

Let X be an irreducible smooth projective curve of genus g ≥ 3 defined over the complex numbers and let Mξ denote the moduli space of stable vector bundles on X of rank n and determinant ξ, where ξ is a fixed line bundle of degree d. If n and d have a common divisor, there is no universal vector bundle on X ×Mξ. We prove that there is a projective bundle on X ×Mξ with the property that its rest...

Let X be a smooth projective curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0. Given a semistable vector bundle E over X , we show that its direct image F∗E under the Frobenius map F of X is again semistable. We deduce a numerical characterization of the stable rank-p vector bundles F∗L, where L is a line bundle over X .

We demonstrate that the notions of derivative representation of a Lie algebra on a vector bundle, of semi-linear representation of a Lie group on a vector bundle, and related concepts, may be understood in terms of representations of Lie algebroids and Lie groupoids, and we indicate how these notions extend to derivative representations of Lie algebroids and semi-linear representations of Lie g...

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikeš, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sec...

On the Kontsevich moduli space of unpointed stable maps to P1 of genus 0 and degree e, there is a tautological vector bundle of rank e − 1. Global sections of tensor powers of this vector bundle arise when considering holomorphic contravariant tensors on Kontsevich spaces of stable maps to more general projective varieties. The computation of the global sections is reduced to an explicit combin...