Related papers: On the Riemann-Hilbert Problems

In this paper we study the deformation problem of pairs consisting of a Riemann surface and a holomorphic line bundle over that surface, and also sections thereof. We emphasize a constructive approach throughout and work and use covering…

We will study a linear first order system, a connection $\db$ problem, on a vector bundle equipped with a connection, over a Riemann surface. We show optimal conditions on the connection forms which allow one to find a holomorphic frame, or…

This note examines the geometry behind the Hamiltonian structure of isomonodromy deformations of connections on vector bundles over Riemann surfaces. The main point is that one should think of an open set of the moduli of pairs $(V,\nabla)$…

On an arbitrary compact Riemann surface, necessary and sufficient conditions are found for the existence of semistable vector bundles with slope between zero and one and a prescribed number of linearly independent holomorphic sections.…

In this paper we provide two ways of constructing complex coordinates on the moduli space of pairs of a Riemann surface and a stable holomorphic vector bundle centred around any such pair. We compute the transformation between the…

We show that the "eigenbundle" (localization bundle) of certain Hilbert modules over bounded symmetric domains of rank r is a "singular" vector bundle (linearly fibrered complex analytic space) which decomposes as a stratified sum of…

We tackle the Riemann-Hilbert problem on the Riemann sphere as stalk-wise logarithmic modifications of the classical R\"ohrl-Deligne vector bundle. We show that the solutions of the Riemann-Hilbert problem are in bijection with some…

The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of…

We construct a pair (E ,F), where E is a holomorphic vector bundle over a compact Riemann surface and F a holomorphic subbundle of E, such that both F and E/F admit holomorphic connections, but E does not.

A holomorphic triple over a compact Riemann surface consists of two holomorphic vector bundles and a holomorphic map between them. After fixing the topological types of the bundles and a real parameter, there exist moduli spaces of stable…

Let $X$ be a connected, compact complex manifold and $S\subset X$ a separating real hypersurface, so that $X$ decomposes as a union of compact complex manifolds with boundary $\bar X^\pm$. Let $\mathcal{M}$ be the moduli space of $S$-framed…

Witten suggested that fixed-point theorems can be derived by the supersymmetric sigma model on a Riemann manifold M with potential term induced from Killing vector on M. One of the well-known fixed-point theorem is the Bott residue formula…

We study existence and uniqueness of a transmission problem in simply connected domains in the plane with data in weighted Lebesgue spaces by first investigating solvability results of a related novel problem associated to a homeomorphism…

We study holomorphic $(n+1)$-chains $E_n\to E_{n-1} \to >... \to E_0$ consisting of holomorphic vector bundles over a compact Riemann surface and homomorphisms between them. A notion of stability depending on $n$ real parameters was…

The vector Riemann-Hilbert problem is analyzed when the entries of its matrix coefficient are meromorphic and almost periodic functions. Three cases for the meromorphic functions, when they have (i) a finite number of poles and zeros…

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle,…

This survey provides an introduction to basic questions and techniques surrounding the topology of the moduli space of stable Higgs bundles on a Riemann surface. Through examples, we demonstrate how the structure of the cohomology ring of…

Into this note we collect topics related to homogeneous vector bundles, elliptic adjoint orbits and so forth.

We identify the holomorphic de Rham complex of the minimal extension of a meromorphic vector bundle with connexion on a compact Riemann surface X with the L^2 complex relative to a suitable metric on the bundle and a complete metric on the…

The Hamiltonian theory of zero-curvature equations with spectral parameter on an arbitrary compact Riemann surface is constructed. It is shown that the equations can be seen as commuting flows of an infinite-dimensional field generalization…