Related papers: Hitchin Hamiltonians in genus 2

Several families of rank-two vector bundles on Hirzebruch surfaces are shown to consist of all very ample, uniform bundles. Under suitable numerical assumptions, the projectivization of these bundles, embedded by their tautological line…

Motivated by the connection to 4d $\mathcal{N}=2$ theories, we study the global behavior of families of tamely-ramified $SL_N$ Hitchin integrable systems as the underlying curve varies over the Deligne-Mumford moduli space of stable pointed…

Let $X$ be a smooth complex projective curve of genus $g$ and let $L$ be a line bundle on $X$ with $\mathrm{deg}\,L>0$. Let $\mathbf{M}$ be the moduli space of semistable rank 2 $L$-twisted Higgs bundles with trivial determinant on $X$. Let…

We prove the Bertram-Feinberg-Mukai conjecture for a generic curve $C$ of genus $g$ and a semistable vector bundle $E$ of rank two and determinant $K$ on $C$, namely we prove the injectivity of the Petri-canonical map $S^2(H^0(E))\to…

We show that the locus of stable rank four vector bundles without theta divisor over a smooth projective curve of genus two is in canonical bijection with the set of theta-characteristics. We give several descriptions of these bundles and…

We prove that given a Hitchin representation in a real split rank 2 group $\mathsf G_0$, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrization of the Hitchin…

We attempt to describe the rank 2 vector bundles on a curve C which are specializations of the trivial bundle. We get a complete classifications when C is Brill-Noether generic, or when it is hyperelliptic; in both cases all limit vector…

In this paper, we construct a stable parabolic Higgs bundle of rank two, which corresponds to the uniformization associated with a conformal hyperbolic metric on a compact Riemann surface $\overline{X}$ with prescribed singularities. This…

The paper gives a quick account of the simplest cases of the Hitchin integrable systems and of the Knizhnik-Zamolodchikov-Bernard connection at genus 0, 1 and 2. In particular, we construct the action-angle variables of the genus 2 Hitchin…

We study the $2k$-Hitchin equations introduced by Ward \cite{Ward 2} from the geometric viewpoint of Higgs bundles. After an introduction on Higgs bundles and $2k$-Hitchin's equations, we review some elementary facts on complex geometry and…

Let $\varphi\colon X\rightarrow Y$ be a degree two Galois cover of smooth curves over a local field $F$ of odd characteristic. Assuming that $Y$ has good reduction, we describe a semi-stability criterion for the curve $X$, using the data of…

Clifford indices of vector bundles on algebraic curves were introduced in a previous paper of the authors. In this paper we study bundles of rank 2 which compute these Clifford indices. This is of particular interest in the light of…

In this paper we count the number of isomorphism classes of geometrically indecomposable quasi-parabolic structures of a given type on a given vector bundle on the projective line over a finite field. We give a conjectural cohomological…

Let $X$ be a smooth irreducible projective curve. Recently, Pauly and Pe\'on-Nieto shows that a vector bundle over $X$ is very stable if and only if the Hitchin map on the vector space of Higgs field on that vector bundle is proper. In this…

We give an algebro-geometric construction of the Hitchin connection, valid also in positive characteristic (with a few exceptions). A key ingredient is a substitute for the Narasimhan-Atiyah-Bott K\"ahler form that realizes the Chern class…

We determine the rational Chow ring of the universal moduli space of rank $2$ semistable bundles over smooth curves of genus $2$, and show that it is generated by certain tautological classes. In the process, we obtain Chow rings of…

We review the notions of symplectic and orthogonal vector bundles over curves, and the connection between principal parts and extensions of vector bundles. We give a criterion for a certain extension of rank 2n to be symplectic or…

For the universal isomonodromic deformation of an irreducible logarithmic rank two connection over a smooth complex projective curve of genus at least two, consider the family of holomorphic vector bundles over curves underlying this…

We prove a gluing theorem for solutions of Hitchin's self-duality equations with logarithmic singularities on a rank-2 vector bundle over a noded Riemann surface representing a boundary point of Teichm\"uller moduli space.

For any V-twisted Higgs bundle on a compact Riemann surface X, where V is a holomorphic vector bundle of rank two on X, there are two associated Higgs bundles on X, twisted by line bundles, which are constructed using a Hecke transformation…