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Jacques Hurtubise

Publications and source records attributed to Jacques Hurtubise.

At least 19 recordsLinked to original sources
arXiv2026

Theta functions for singular curves

Let $X$ be an irreducible singular Riemann surface, with desingularisation $\widetilde X$. The generalised Jacobian $J(X)$ of $X$ fibers over the Jacobian $J(\widetilde{X})$ of $\widetilde X$, and there is an Abel map $A$ of $\widetilde X$ to $J(X)$, lifting the Abel map to $J(\widetilde X)$. We build a theta function on a compactification of the generalised Jacobian $J(X)$ (giving a section of a suitable positive line bundle). The translation action on $J(X)$ then yields all line bundles of that degree, and the translates of the theta function, restricted to $A(\widetilde X)$, give a ``universal section'' of the line bundles of that degree over $X$. This extends to the singular case a classical result of Riemann.

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arXiv2025

Torsors over moduli spaces of vector bundles over curves of fixed determinant

Let ${\mathcal M}$ be a moduli space of stable vector bundles of rank $r$ and determinant $ξ$ on a compact Riemann surface $X$. Fix a semistable holomorphic vector bundle $F$ on $X$ such that $χ(E\otimes F)= 0$ for $E \in \mathcal M$. Then any $E\in \mathcal M$ with $H^0(X, E\otimes F) = 0 = H^1(X, E\otimes F)$ has a natural holomorphic projective connection. The moduli space of pairs $(E,\, \nabla)$, where $E\, \in\, \mathcal M$ and $\nabla$ is a holomorphic projective connection on $E$, is an algebraic $T^*{\mathcal M}$--torsor on $\mathcal M$. We identify this $T^*{\mathcal M}$--torsor on $\mathcal M$ with the $T^*{\mathcal M}$--torsor given by the sheaf of connections on an ample line bundle over $\mathcal M$.

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arXiv2023

Moduli spaces of vector bundles on a curve and opers

Let $X$ be a compact connected Riemann surface of genus $g$, with $g\, \geq\,2$, and let $ξ$ be a holomorphic line bundle on $X$ with $ξ^{\otimes 2}\,=\, {\mathcal O}_X$. Fix a theta characteristic $\mathbb L$ on $X$. Let ${\mathcal M}_X(r,ξ)$ be the moduli space of stable vector bundles $E$ on $X$ of rank $r$ such that $\bigwedge^r E\,=\, ξ$ and $H^0(X,\, E\otimes{\mathbb L})\,=\, 0$. Consider the quotient of ${\mathcal M}_X(r,ξ)$ by the involution given by $E\, \longmapsto\, E^*$. We construct an algebraic morphism from this quotient to the moduli space of ${\rm SL}(r,{\mathbb C})$ opers on $X$. Since $\dim {\mathcal M}_X(r,ξ)$ coincides with the dimension of the moduli space of ${\rm SL}(r,{\mathbb C})$ opers, it is natural to ask about the injectivity and surjectivity of this map.

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arXiv2023

Framed Parabolic sheaves on a trinion

We consider for structure groups ${\rm SU}(n)\,\subset\, {\rm SL}(n,\mathbb C)$ a densely defined toric structure on the moduli of framed parabolic sheaves on a three-punctured sphere, which degenerates to an actual toric structure. In combination with previous degeneration results, these extend to similar moduli for arbitrary Riemann surfaces.

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arXiv2023

Degenerations of Bundle Moduli

Over a family $\mathbb X$ of genus $g$ complete curves, which gives the degeneration of a smooth curve into one with nodal singularities, we build a moduli space which is the moduli space of ${\rm SL}(n, \mathbb C)$ bundles over the generic smooth curve $X_t$ in the family, and is a moduli space of bundles equipped with extra structure at the nodes for the nodal curves in the family. This moduli space is a quotient by $(\mathbb C^*)^s$ of a moduli space on the desingularisation. Taking a "maximal" degeneration of the curve into a nodal curve built from the glueing of three-pointed spheres, we obtain a degeneration of the moduli space of bundles into a $(\mathbb C^*)^{(3g-3)(n-1)}$-quotient of a $(2g-2)$-th power of a space associated to the three-pointed sphere. Via the Narasimhan-Seshadri theorem, the moduli of bundles on the smooth curve is a space of representations of the fundamental group into ${\rm SU}(n)$ (the "symplectic picture"). We obtain the degenerations also in this symplectic context, in a way that is compatible with the holomorphic degeneration, so that our limit space is also a $(S^1)^{(3g-3)(n-1)}$ symplectic quotient of a $(2g-2)$-th power of a space associated to the three-pointed sphere.

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arXiv2022

Poisson maps between character varieties: gluing and capping

Let G be a compact Lie group or a complex reductive affine algebraic group. We explore induced mappings between G-character varieties of surface groups by mappings between corresponding surfaces. It is shown that these mappings are generally Poisson. We also given an effective algorithm to compute the Poisson bi-vectors when G=SL(2,C). We demonstrate this algorithm by explicitly calculating the Poisson bi-vector for the 5-holed sphere, the first example for an Euler characteristic -3 surface.

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arXiv2021

Vector bundles and connections on Riemann surfaces with projective structure

Let ${\mathcal B}_g(r)$ be the moduli space of triples of the form $(X,\, K^{1/2}_X,\, F)$, where $X$ is a compact connected Riemann surface of genus $g$, with $g\, \geq\, 2$, $K^{1/2}_X$ is a theta characteristic on $X$, and $F$ is a stable vector bundle on $X$ of rank $r$ and degree zero. We construct a $T^*{\mathcal B}_g(r)$--torsor ${\mathcal H}_g(r)$ over ${\mathcal B}_g(r)$. This generalizes on the one hand the torsor over the moduli space of stable vector bundles of rank $r$, on a fixed Riemann surface $Y$, given by the moduli space of holomorphic connections on the stable vector bundles of rank $r$ on $Y$, and on the other hand the torsor over the moduli space of Riemann surfaces given by the moduli space of Riemann surfaces with a projective structure. It is shown that ${\mathcal H}_g(r)$ has a holomorphic symplectic structure compatible with the $T^*{\mathcal B}_g(r)$--torsor structure. We also describe ${\mathcal H}_g(r)$ in terms of the second order matrix valued differential operators. It is shown that ${\mathcal H}_g(r)$ is identified with the $T^*{\mathcal B}_g(r)$--torsor given by the sheaf of holomorphic connections on the theta line bundle over ${\mathcal B}_g(r)$.

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arXiv2021

A canonical connection on bundles on Riemann surfaces and Quillen connection on the theta bundle

We investigate the symplectic geometric and differential geometric aspects of the moduli space of connections on a compact Riemann surface $X$. Fix a theta characteristic $K^{1/2}_X$ on $X$; it defines a theta divisor on the moduli space ${\mathcal M}$ of stable vector bundles on $X$ of rank $r$ degree zero. Given a vector bundle $E \in {\mathcal M}$ lying outside the theta divisor, we construct a natural holomorphic connection on $E$ that depends holomorphically on $E$. Using this holomorphic connection, we construct a canonical holomorphic isomorphism between the following two: \begin{enumerate} \item the moduli space $\mathcal C$ of pairs $(E, D)$, where $E\in {\mathcal M}$ and $D$ is a holomorphic connection on $E$, and \item the space ${\rm Conn}(Θ)$ given by the sheaf of holomorphic connections on the line bundle on $\mathcal M$ associated to the theta divisor. \end{enumerate} The above isomorphism between $\mathcal C$ and ${\rm Conn}(Θ)$ is symplectic structure preserving, and it moves holomorphically as $X$ runs over a holomorphic family of Riemann surfaces.

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arXiv2020

Meromorphic connections, determinant line bundles and the Tyurin parametrization

We develop a holomorphic equivalence between on one hand the space of pairs (stable bundle, flat connection on the bundle) and the "sheaf of holomorphic connections" (the sheaf of splittings of the one-jet sequence) for the determinant (Quillen) line bundle over the moduli space of vector bundles on a compact connected Riemann surface. This equivalence is shown to be holomorphically symplectic. The equivalences, both holomorphic and symplectic, seem to be quite general, in that they extend to other general families of holomorphic bundles and holomorphic connections, in particular those arising from "Tyurin families" of stable bundles over the surface. These families generalize the Tyurin parametrization of stable vector bundles $E$ over a compact connected Riemann surface, and one can build above them spaces of (equivalence classes of) connections, which are again symplectic. These spaces are also symplectically biholomorphically equivalent to the sheaf of connections for the determinant bundle over the Tyurin family. The last portion of the paper shows how this extends to moduli of framed bundles.

math.AG↗
arXiv2020

Instantons and Bows for the Classical Groups

The construction of Atiyah, Drinfeld, Hitchin, and Manin [ADHM78] provided complete description of all instantons on Euclidean four-space. It was extended by Kronheimer and Nakajima to instantons on ALE spaces, resolutions of orbifolds $\mathbb{R}^4/Γ$ by a finite subgroup $Γ\subset SU(2).$ We consider a similar classification, in the holomorphic context, of instantons on some of the next spaces in the hierarchy, the ALF multi-Taub-NUT manifolds, showing how they tie in to the bow solutions to Nahm's equations [Che09] via the Nahm correspondence. Recently in [Nak18a] and [NT17], based on [Nak03], Nakajima and Takayama constructed the Coulomb branch of the moduli space of vacua of a quiver gauge theory, tying them to the same space of bow solutions. One can view our construction as describing the same manifold as the Higgs branch of the mirror gauge theory [COS11]. Our construction also yields the monad construction of holomorphic instanton bundles on the multi-Taub-NUT space for any classical compact Lie structure group.

math.DG↗
arXiv2019

Monads for Instantons and Bows

Instantons on the Taub-NUT space are related to `bow solutions' via a generalization of the ADHM-Nahm transform. Both are related to complex geometry, either via the twistor transform or via the Kobayashi-Hitchin correspondence. We explore various aspects of this complex geometry, exhibiting equivalences. For both the instanton and the bow solution we produce two monads encoding each of them respectively. Identifying these monads we establish the one-to-one correspondence between the instanton and the bow solution.

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arXiv2019

Isomonodromic deformations of logarithmic connections and stable parabolic vector bundles

We consider irreducible logarithmic connections $(E,\,δ)$ over compact Riemann surfaces $X$ of genus at least two. The underlying vector bundle $E$ inherits a natural parabolic structure over the singular locus of the connection $δ$; the parabolic structure is given by the residues of $δ$. We prove that for the universal isomonodromic deformation of the triple $(X,\,E,\,δ)$, the parabolic vector bundle corresponding to a generic parameter in the Teichmüller space is parabolically stable. In the case of parabolic vector bundles of rank two, the general parabolic vector bundle is even parabolically very stable.

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arXiv2018

Principal co-Higgs bundles on $\mathbb{P}^1$

For complex connected, reductive, affine, algebraic groups $G$, we give a Lie-theoretic characterization of the semistability of principal $G$-co-Higgs bundles on the complex projective line $\mathbb{P}^1$ in terms of the simple roots of a Borel subgroup of $G$. We describe a stratification of the moduli space in terms of the Harder-Narasimhan type of the underlying bundle.

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arXiv2018

Higgs bundles, branes and Langlands duality

Given a compact Riemann surface $X$ and a complex reductive Lie group $G$ equipped with real structures, we define antiholomorphic involutions on the moduli space of $G$-Higgs bundles over $X$. We investigate how the various components of the fixed point locus match up, as one passes from $G$ to its Langlands dual $^LG$. Special attention is given to the case $G=SL(2,\C)$ and $^LG=PGL(2,\C)$.

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arXiv2017

Spatially periodic instantons: Nahm transform and moduli

This paper establishes that the Nahm transform sending spatially periodic instantons (instantons on the product of the real line and a three-torus) to singular monopoles on the dual three-torus is indeed a bijection as suggested by the heuristic. In the process, we show how the Nahm transform intertwines to a Fourier-Mukai transform via Kobayashi-Hitchin correspondences. We also prove existence and non-existence results.

math.DG↗
arXiv2017

Spectral curves for the triple reduced product of coadjoint orbits for SU(3)

We give an identification of the triple reduced product of three coadjoint orbits in SU(3) with a space of Hitchin pairs over a genus 0 curve with three punctures, where the residues of the Higgs field at the punctures are constrained to lie in fixed coadjoint orbits. Using spectral curves for the corresponding Hitchin system, we identify the moment map for a Hamiltonian circle action on the reduced product. Finally, we make use of results of Adams, Harnad, and Hurtubise to find Darboux coordinates and a differential equation for the Hamiltonian.

math.AG↗
arXiv2017

Isomonodromic deformations and very stable vector bundles of rank two

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 universal deformation. In a previous work we proved that the vector bundle corresponding to a general parameter of this family is stable. Here we prove that the vector bundle corresponding to a general parameter is in fact very stable (it does not admit any nonzero nilpotent Higgs field).

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