Related papers: On the Brill-Noether Problem for Vector Bundles

The Abelian Higgs model on a compact Riemann surface \Sigma of genus g is considered. We show that for g > 1 the Bogomolny equations for multi-vortices at critical coupling can be obtained as compatibility conditions of two linear equations…

In this paper, we consider the existence of approximate Hermitian-Einstein structure and the semi-stability on Higgs bundles over compact Gauduchon manifolds. By using the continuity method, we show that they are equivalent.

Here we consider the product of varieties with $n$-blocks collections . We give some cohomological splitting conditions for rank 2 bundles. A cohomological characterization for vector bundles is also provided. The tools are Beilinson's type…

In an influential 2008 paper, Baker proposed a number of conjectures relating the divisor theory of algebraic curves with an analogous combinatorial theory on finite graphs. In this note, we examine Baker's Brill--Noether existence…

We give a formalism for constructing hidden sector bundles as extensions of sums of line bundles in heterotic $M$-theory. Although this construction is generic, we present it within the context of the specific Schoen threefold that leads to…

In this work, we study topological properties of surface bundles, with an emphasis on surface bundles with a spin structure. We develop a criterion to decide whether a given manifold bundle has a spin structure and specialize it to surface…

We consider stable minimal surfaces of genus 1 in Euclidean space and in Riemannian manifolds. Under the condition of covering stability (all finite covers are stable) we show that a genus 1 finite total curvature minimal surface in…

Given a tuple of holomorphic differentials on a Riemann surface, one can define a Higgs bundle in the Hitchin section and a natural symmetric pairing of the Higgs bundle. We study whether a Higgs bundle of rank 3 in the Hitchin section has…

We extend the notion of the spectrum of semistable rank two bundles (initially constructed on ${\bf P}^3$ by Barth and Elencwajg) to a certain class of compact complex algebraic threefolds.

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…

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…

A new class of Higgs bundles is introduced in a natural setting. Existence and nonexistence results for Higgs-Hermitian-Yang-Mills metrics are proved.

Let $(M,g)$ be a smooth compact Riemannian surface with no boundary. Given a smooth vector field $V$ with finitely many zeroes on $M$, we study the distribution of the number of tangencies to $V$ of the nodal components of random…

Let $X$ be a K3 surface with Picard group $\mathrm{Pic}(X)\cong\mathbb{Z} H$ such that $H^2=2n$. Let $M_{H}(\mathbf{v})$ be the moduli space of Gieseker semistable sheaves on $X$ with Mukai vector $\mathbf{v}$. We say that $\mathbf{v}$…

We study Brill-Noether existence on a finite graph using methods from polyhedral geometry and lattices. We start by formulating analogues of the Brill-Noether conjectures (both the existence and non-existence parts) for…

We consider circle bundles over compact three-manifolds with symplectic total spaces. We show that the base of such a space must be irreducible or the product of the two-sphere with the circle. We then deduce that such a bundle admits a…

We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact K\"ahler manifolds and we present…

Clifford indices for semistable vector bundles on a smooth projective curve of genus at least 4 were defined in a previous paper of the authors. The present paper studies bundles which compute these Clifford indices. We show that under…

The problem of a spinless particle subject to a general mixing of vector and scalar screened Coulomb potentials in a two-dimensional world is analyzed and its bounded solutions are found. Some unusual results, including the existence of a…

This article studies the harmonicity of vector fields on Riemannian manifolds, viewed as maps into the tangent bundle equipped with a family of Riemannian metrics. Geometric and topological rigidity conditions are obtained, especially for…