Interpolation Problems for Vector Bundles on Algebraic Curves

In this paper we study meromorphic maps between vector bundles on a Riemann surface. We are mainly interested in stable vector bundles. For a huge number of numerical data we prove the existence of a meromorphic map between two vector bundles with a prescribed number of zeroes and a prescribed number of poles.     

Exact Sequences of Semistable Vector Bundles on Algebraic Curves

Let X be a smooth complex projective curve of genus g 1. Ifg 2, then assume further that X is either bielliptic or withgeneral moduli. Fix integers r, s, a, b with r > 1, s >1 and as br. Here we prove the existence of an exact sequence [formula] of semistable vector bundles on X with rk(H) = r, rk(Q) = s,deg(H) = a and deg(Q) = b. 1991 Mathematics Subject Classification14H60.     

Moduli of Vector Bundles on Curves in Positive Characteristics

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.     

Vector Bundles Near Negative Curves: Moduli and Local Euler Characteristic

We study moduli of vector bundles on a two-dimensional neighbourhood Z _k of an irreducible curve ? ? ?¹ with ?² = ?k and give an explicit construction of their moduli stacks. For the case of instanton bundles, we stratify the stacks and construct moduli spaces. We give sharp bounds for the local holomorphic Euler characteristic of bundles on Z _k and prove existence of families of bundles with prescribed numerical invariants. Our numerical calculations are performed using a Macaulay 2 algorithm, which is available for download at http://www.maths.ed.ac.uk/~s0571100/Instanton/.     

A Universal Construction for Moduli Spaces of Decorated Vector Bundles over Curves

Let $X$ be a smooth projective curve over the field of complex numbers, and fix a homogeneous representation $\rho\colon \mathop{\rm GL}(r)\rightarrow \mathop{\rm GL}(V)$. Then one can associate to every vector bundle $E$ of rank $r$ over $X$ a vector bundle $E_\rho$ with fibre $V$. We would like to study triples $(E,L,\phi)$ where $E$ is a vector bundle of rank $r$ over $X$, $L$ is a line bundle over $X$, and $\phi\colon E_\rho\rightarrow L$ is a nontrivial homomorphism. This setup comprises well known objects such as framed vector bundles, Higgs bundles, and conic bundles. In this paper, we will formulate a general (parameter dependent) semistability concept for such triples, which generalizes the classical Hilbert--Mumford criterion, and we establish the existence of moduli spaces for the semistable objects. In the examples which have been studied so far, our semistability concept reproduces the known ones. Therefore, our results give in particular a unified construction for many moduli spaces considered in the literature.     

Real Abelian Varieties with Many Line Bundles

Let X and Y be affine nonsingular real algebraic varieties.A general problem in real algebraic geometry is to try to decidewhen a continuous map f: X Y can be approximated by regularmaps in the space of c⁰ mappings from X to Y, equipped withthe c⁰ topology. This paper solves this problem when X is theconnected component containing the origin of the real part ofa complex Abelian variety and Y is the standard 2-dimensionalsphere.     

M.P. Appell's Function and Vector Bundles of Rank 2 on Elliptic Curves

This paper is devoted to the function introduced by M.P. Appell in connection with decomposition of elliptic functions of the third kind into simple elements. We show that this function is related to global sections of rank-2 vector bundles on elliptic curves. We derive analogues of theta-identities for this function and prove the divisibility property for the action of the modular group, that should be considered as a replacement of the functional equation.     

Genus Two Curves with Many Elliptic Subcovers

We determine all genus 2 curves, defined over ?, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in ?₂. For each component, we find a rational parametrization and construct the equation of the corresponding genus 2 curve and its elliptic subcovers in terms of the parameterization. Such families of genus 2 curves are determined for the first time. Furthermore, we prove that there are only finitely many genus 2 curves (up to ?-isomorphism) defined over ?, which have degree 2 and 3 elliptic subcovers also defined over ?.     

On Arcs and Curves with Many Automorphisms

We construct highly symmetric arcs by using highly symmetric curves: the Klein quartic which is the most symmetric non-singular curve of degree 4, and the Wiman sextic which is shown to be the unique A₆-invariant curve of degree 6. The set of flexes of the Klein quartic is a 24-arc with automorphism group PSL(2, 7), while the set of flexes of the Wiman sextic is a 72-arc with automorphism group PSL(2, 9) A₆.This research was carried out with the support of the Italian MIUR (Progetto Strutture geometriche, combinatoria e loro applicazioni) and of GNSAGA.     

Vector Bundles on Certain Non-algebraic Surfaces

It is well-known that every holomorphic vector bundle is filtrable on a projective algebraic     

Convergence of Vector Bundles with Metrics of Sasaki-Type

If a sequence of Riemannian manifolds, X _i , converges in the pointed Gromov–Hausdorff sense to a limit space, X _∞, and if E _i are vector bundles over X _i endowed with metrics of Sasaki-type with a uniform upper bound on rank, then a subsequence of the E _i converges in the pointed Gromov–Hausdorff sense to a metric space, E _∞. The projection maps π _i converge to a limit submetry π _∞ and the fibers converge to its fibers; the latter may no longer be vector spaces but are homeomorphic to \(\mathbb {R}^{k}/G\) , where G, henceforth called the wane group, is a closed subgroup of O(k) that depends on the basepoint and that is defined using the holonomy groups on the vector bundles. The norms μ _i =∥?∥ _i converge to a map μ _∞ compatible with the rescaling in \(\mathbb {R}^{k}/G\) and the \(\mathbb {R}\) -action on E _i converges to an \(\mathbb {R}\) -action on E _∞ compatible with the limiting norm. A natural notion of parallelism is given to the limiting spaces by considering curves whose length is unchanged under the projection. The class of such curves is invariant under the \(\mathbb {R}\) -action and each such curve preserves norms. The existence of parallel translation along rectifiable curves with arbitrary initial conditions is also exhibited. Also, necessary conditions for uniqueness of parallel translates are given in terms of the wane groups. In the special case when the sequence of vector bundles has a uniform lower bound on holonomy radius (as in a sequence of collapsing flat tori to a circle), the limit fibers are vector spaces. Under the opposite extreme, e.g., when a single compact n-dimensional manifold is rescaled to a point, the limit fiber is \(\mathbb {R}^{n}/H\) where H is the closure of the holonomy group of the compact manifold considered. Both these examples have uniqueness of parallel translates. However, examples for non-uniqueness are also produced by looking at isolated conical singularities.     

Echelon Modifications of Vector Bundles

We study a filtered generalization of the operation of elementary modification of vector bundles. The generalization is motivated by applications to the degeneration theory of linear systems.     

Remarks on Normal Generation of Special Line Bundles on Algebraic Curves

Let L be a very ample line bundle with h ¹(L) ≥2 on a curve of genus g. We prove that L is normally generated if deg(L) ≥2g ? 1 ? 4h ¹(L) for large enough genus g.     

Homogeneous Vector Bundles and Koszul Algebras

Let G be a reductive algebraic group defined over an algebraically closed field of characteristic zero and let P be a parabolic subgroup of G. We consider the category of homogeneous vector bundles over the flag variety G/P. This category is equivalent to a category of representations of a certain infinite quiver with relations by a generalisation of a result in [BK]. We prove that both categories are Koszul precisely when the unipotent radical P_u of P is abelian.     

η-Invariant and Flat Vector Bundles

We present an alternate definition of the mod Z component of the AtiyahPatodi-Singer η invariant associated to (not necessary unitary) fiat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah, Parodi and Singer.