arXiv subjects

Jacques Hurtubise

Publications and source records attributed to Jacques Hurtubise.

At least 37 records · Page 2Linked to original sources

arXiv2016

Isomonodromic deformations of irregular connections and stability of bundles

Let $G$ be a reductive affine algebraic group defined over $\mathbb C$, and let $\nabla_0$ be a meromorphic $G$-connection on a holomorphic $G$-bundle $E_0$, over a smooth complex curve $X_0$, with polar locus $P_0 \subset X_0$. We assume that $\nabla_0$ is irreducible in the sense that it does not factor through some proper parabolic subgroup of $G$. We consider the universal isomonodromic deformation $(E_t\to X_t, \nabla_t, P_t)_{t\in \mathcal{T}}$ of $(E_0\to X_0, \nabla_0, P_0)$, where $\mathcal{T}$ is a certain quotient of a certain framed Teichmüller space we describe. We show that if the genus $g$ of $X_0$ satisfies $g\geq 2$, then for a general parameter $t\in \mathcal{T}$, the $G$-bundle $E_t\to X_t$ is stable. For $g\geq 1$, we are able to show that for a general parameter $t\in \mathcal{T}$, the $G$-bundle $E_t\to X_t$ is semistable.

math.AG↗

arXiv2016

Moduli space for generic unfolded differential linear systems

In this paper, we identify the moduli space for germs of generic unfoldings of nonresonant linear differential systems with an irregular singularity of Poincaré rank $k$ at the origin, under analytic equivalence. The modulus of a given family was determined in \cite{HLR}: it comprises a formal part depending analytically on the parameters, and an analytic part given by unfoldings of the Stokes matrices. These unfoldings are given on "Douady-Sentenac" (DS) domains in the parameter space covering the generic values of the parameters corresponding to Fuchsian singular points. Here we identify exactly which moduli can be realized. A necessary condition on the analytic part, called compatibility condition, is saying that the unfoldings define the same monodromy group (up to conjugacy) for the different presentations of the modulus on the intersections of DS domains. With the additional requirement that the corresponding cocycle is trivial and good limit behavior at some boundary points of the DS domains, this condition becomes sufficient. In particular we show that any modulus can be realized by a $k$-parameter family of systems of rational linear differential equations over $\mathbb C\mathbb P^1$ with $k+1$, $k+2$ or $k+3$ singular points (with multiplicities). Under the generic condition of irreducibility, there are precisely $k+2$ singular points which are Fuchsian as soon as simple. This in turn implies that any unfolding of an irregular singularity of Poincaré rank $k$ is analytically equivalent to a rational system of the form $y'=\frac{A(x)}{p_ε(x)}\cdot y$, with $A(x)$ polynomial of degree at most $k$ and $p_ε(x)$ is the generic unfolding of the polynomial $x^{k+1}$.

math.DS↗

arXiv2015

Isomonodromic deformations of logarithmic connections and stability

Let X_0 be a compact connected Riemann surface of genus g with D_0\subset X_0 an ordered subset of cardinality n, and let E_G be a holomorphic principal G-bundle on X_0, where G is a complex reductive affine algebraic group, that admits a logarithmic connection \nabla_0 with polar divisor D_0. Let (\cal{E}_G, \nabla) be the universal isomonodromic deformation of (E_G,\nabla_0) over the universal Teichmüller curve (\cal{X}, \cal{D})\rightarrow {Teich}_{g,n}, where {Teich}_{g,n} is the Teichmüller space for genus g Riemann surfaces with n-marked points. We prove the following: Assume that g>1 and n= 0. Then there is a closed complex analytic subset \cal{Y} \subset {Teich}_{(g,n)}, of codimension at least $g$, such that for any t\in {Teich}_{(g,n)} \setminus \mathcal{Y}, the principal G-bundle \cal{E}_G\vert_{{\cal X}_t} is semistable, where {\cal X}_t is the compact Riemann surface over $t$. Assume that g>0, and if g= 1, then n >0. Also, assume that the monodromy representation for \nabla_0 does not factor through some proper parabolic subgroup of G. Then there is a closed complex analytic subset $\cal{Y}' \subset {Teich}_{(g,n)}, of codimension at least g, such that for any t\in {Teich}_{(g,n)} \setminus \cal{Y}', the principal G-bundle $\cal{E}_G\vert_{{\cal X}_t}$ is semistable. Assume that g>1. Assume that the monodromy representation for \nabla_0 does not factor through some proper parabolic subgroup of G. Then there is a closed complex analytic subset \cal{Y}" \subset {Teich}_{(g,n)}, of codimension at least g-1, such that for any t\in {Teich}_{(g,n)} \setminus \cal{Y}', the principal G-bundle \cal{E}_G\vert_{{\cal X}_t} is stable.

math.AG↗

arXiv2015

Pseudo-real principal $G$-bundles over a real curve

We consider stable and semistable principal bundles over a smooth projective real algebraic curve, equipped with a real or pseudo-real structure in the sense of Atiyah. After fixing suitable topological invariants, one can build a suitable gauge theory, and show that the resulting moduli spaces of pseudo-real bundles are connected. This in turn allows one to describe the various fixed point varieties on the complex moduli spaces under the action of the real involutions on the curve and the structure group.

math.AG↗

arXiv2015

A symplectic analog of the Quot scheme

We construct a symplectic analog of the Quot scheme that parametrizes the torsion quotients of a trivial vector bundle over a compact Riemann surface. Some of its properties are investigated.

math.AG↗

arXiv2015

Brauer Groups of Quot Schemes

Let $X$ be an irreducible smooth complex projective curve. Let ${\mathcal Q}(r,d)$ be the Quot scheme parametrizing all coherent subsheaves of ${\mathcal O}^{\oplus r}_X$ of rank $r$ and degree $-d$. There are natural morphisms ${\mathcal Q}(r,d) \longrightarrow \text{Sym}^d(X)$ and $\text{Sym}^d(X) \longrightarrow \text{Pic}^d(X)$. We prove that both these morphisms induce isomorphism of Brauer groups if $d \geq 2$. Consequently, the Brauer group of ${\mathcal Q}(r,d)$ is identified with the Brauer group of $\text{Pic}^d(X)$ if $d \geq 2$.

math.AG↗

arXiv2014

Monopoles on Sasakian Three-folds

We consider monopoles with singularities of Dirac type on quasiregular Sasakian three-folds fibering over a compact Riemann surface $Σ$, for example the Hopf fibration $S^3\longrightarrow S^2$. We show that these correspond to holomorphic objects on $Σ$, which we call twisted bundle triples. These are somewhat similar to Murray's bundle gerbes. A spectral curve construction allows us to classify these structures, and, conjecturally, monopoles.

math-ph↗

arXiv2014

A generalized Quot scheme and meromorphic vortices

Let X be a compact connected Riemann surface. Fix a positive integer r and two nonnegative integers d_p and d_z. Consider all pairs of the form (F, f), where F is a holomorphic vector bundle on X of rank r and degree d_z-d_p, and f : {\mathcal O}^{\oplus r}_X \rightarrow F is a meromorphic homomorphism which an isomorphism outside a finite subset of X and has pole (respectively, zero) of total degree d_p (respectively, d_z). Two such pairs $(F_1, f_1) and $(F_2, f_2) are called isomorphic if there is a holomorphic isomorphism of F_1 with F_2 over X that takes f_1 to f_2. We construct a natural compactification of the moduli space equivalence classes pairs of the above type. The Poincaré polynomial of this compactification is computed.

math.AG↗

arXiv2014

Poisson surfaces and algebraically completely integrable systems

One can associate to many of the well known algebraically integrable systems of Jacobians (generalized Hitchin systems, Sklyanin) a ruled surface which encodes much of its geometry. If one looks at the classification of such surfaces, there is one case of a ruled surface that does not seem to be covered. This is the case of projective bundle associated to the first jet bundle of a topologically nontrivial line bundle. We give the integrable system corresponding to this surface; it turns out to be a deformation of the Hitchin system.

math.AG↗

arXiv2013

Grassmannian framed bundles and generalized parabolic structures

We build compact moduli spaces of Grassmannian framed bundles over a Riemann surface, essentially replacing a group by its bi-invariant compactification. We do this both in the algebraic and symplectic settings, and prove a Hitchin-Kobayashi correspondence between the two. The spaces are universal spaces for parabolic bundles, and the reduction to parabolic bundles commutes with the correspondence. An analogous correspondence is outlined for the generalized parabolic bundles of Bhosle.

math.AG↗

arXiv2012

Automorphisms of the Quot schemes associated to compact Riemann surfaces

Let X be a compact connected Riemann surface of genus at least two. Fix positive integers r and d. Let Q denote the Quot scheme that parametrizes the torsion quotients of {\mathcal O}^{\oplus r}_X of degree d. This Q is also the moduli space of vortices for the standard action of U(r) on {\mathbb C}^r. The group \text{PGL}(r, {\mathbb C}) acts on Q via the action of $\text{GL}(r, {\mathbb C})$ on ${\mathcal O}^{\oplus r}_X$. We prove that this subgroup $\text{PGL}(r, {\mathbb C})$ is the connected component, containing the identity element, of the holomorphic automorphism group Aut(\mathcal Q). As an application of it, we prove that the isomorphism class of the complex manifold Q uniquely determines the isomorphism class of the Riemann surface X.

math.AG↗

arXiv2012

Rank one connections on abelian varieties, II

Given a holomorphic line bundle $L$ on a compact complex torus $A$, there are two naturally associated holomorphic $Ω_A$--torsors over $A$: one is constructed from the Atiyah exact sequence for $L$, and the other is constructed using the line bundle $(p^*_1 L^*)\otimes (α^*L)$, where $α$ is the addition map on $A\times A$, and $p_1$ is the projection of $A\times A$ to the first factor. In \cite{BHR}, it was shown that these two torsors are isomorphic. The aim here is to produce a canonical isomorphism between them through an explicit construction.

math.AG↗

arXiv2012

Pseudo-real principal Higgs bundles on compact Kaehler manifolds

Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form $σ_G$. We define pseudo-real principal $G$--bundles on $X$; these are generalizations of real algebraic principal $G$--bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal $G$--bundles. Their relationships with the usual stable, semistable and polystable principal $G$--bundles are investigated. We then prove that the following Donaldson--Uhlenbeck--Yau type correspondence holds: a pseudo-real principal $G$--bundle admits a compatible Einstein-Hermitian connection if and only if it is polystable. A bijection between the following two sets is established: 1) The isomorphism classes of polystable pseudo-real principal $G$--bundles such that all the rational characteristic classes of the underlying topological principal $G$--bundle vanish. 2) The equivalence classes of twisted representations of the extended fundamental group of $X$ in a $σ_G$--invariant maximal compact subgroup of $G$. (The twisted representations are defined using the central element in the definition of a pseudo-real principal $G$--bundle.) All these results are also generalized to the pseudo-real Higgs $G$--bundle.

math.AG↗

arXiv2012

Principal bundles over a real algebraic curve

Let X be a compact connected Riemann surface equipped with an anti-holomorphic involution σ. Let G be a connected complex reductive affine algebraic group, and let σ_G be a real form of G. We consider holomorphic principal G-bundles on X satisfying compatibility conditions with respect to σand σ_G. We prove that the points defined over $\mathbb R$ of the smooth locus of a moduli space of principal G-bundles on X are precisely these objects, under the assumption that {\rm genus}(X) > 2. Stable, semistable and polystable bundles are defined in this context. Relationship between any of these properties and the corresponding property of the underlying holomorphic principal G-bundle is explored. A bijective correspondence between unitary representations and polystable objects is established.

math.AG↗

arXiv2012

Real projective structures on a real curve

Given a compact connected Riemann surface $X$ equipped with an antiholomorphic involution $τ$, we consider the projective structures on $X$ satisfying a compatibility condition with respect to $τ$. For a projective structure $P$ on $X$, there are holomorphic connections and holomorphic differential operators on $X$ that are constructed using $P$. When the projective structure $P$ is compatible with $τ$, the relationships between $τ$ and the holomorphic connections, or the differential operators, associated to $P$ are investigated. The moduli space of projective structures on a compact oriented $C^\infty$ surface of genus $g\, \geq\, 2$ has a natural holomorphic symplectic structure. It is known that this holomorphic symplectic manifold is isomorphic to the holomorphic symplectic manifold defined by the total space of the holomorphic cotangent bundle of the Teichmüller space ${\mathcal T}_g$ equipped with the Liouville symplectic form. We show that there is an isomorphism between these two holomorphic symplectic manifolds that is compatible with $τ$.

math.AG↗

arXiv2011

Rank one connections on abelian varieties

Let A be a complex abelian variety. The moduli space ${\mathcal M}_C$ of rank one algebraic connections on $A$ is a principal bundle over the dual abelian variety $A^\vee=\text{Pic}^0(A)$ for the group $H^0(A, Ω^1_A)$. Take any line bundle $L$ on $A^\vee$; let ${\mathcal C}(L)$ be the algebraic principal $H^0(A^\vee, Ω^1_{A^\vee})$-bundle over $A^\vee$ given by the sheaf of connections on $L$. The line bundle $L$ produces a homomorphism $H^0(A, Ω^1_A) \rightarrow H^0(A^\vee,\, Ω^1_{A^\vee})$. We prove that ${\mathcal C}(L)$ is isomorphic to the principal $H^0(A^\vee, Ω^1_{A^\vee})$-bundle obtained by extending the structure group of the principal $H^0(A,\, Ω^1_A)$-bundle ${\mathcal M}_C$ using this homomorphism given by $L$. We compute the ring of algebraic functions on ${\mathcal C}(L)$.

math.AG↗

arXiv2010

Universal vector bundle over the reals

Let X_R be a geometrically irreducible smooth projective curve, defined over R, such that X_R does not have any real points. Let X= X_R\times_R C be the complex curve. We show that there is a universal real algebraic line bundle over X_R x Pic^d(X_R)$ if and only if $χ(L)$ is odd for L in Pic^d(X_R)$. There is a universal quaternionic algebraic line bundle over X x Pic^d(X) if and only if the degree d is odd. Take integers r and d such that r > 1, and d is coprime to r. Let M_{X_R}(r,d) (respectively, M_X(r,d)$) be the moduli space of stable vector bundles over X_R (respectively, X) of rank r and degree d. We prove that there is a universal real algebraic vector bundle over X_R x M_{X_R}(r,d) if and only if χ(E) is odd for E in M_{X_R}(r,d). There is a universal quaternionic vector bundle over X x M_X(r,d) if and only if the degree d is odd. The cases where X_R is geometrically reducible or X_R has real points are also investigated.

math.AG↗

arXiv2009

Singular Hermitian-Einstein monopoles on the product of a circle and a Riemann surface

In this paper, the moduli space of singular unitary Hermitian--Einstein monopoles on the product of a circle and a Riemann surface is shown to correspond to a moduli space of stable pairs on the Riemann surface. These pairs consist of a holomorphic vector bundle on the surface and a meromorphic automorphism of the bundle. The singularities of this automorphism correspond to the singularities of the singular monopole. We then consider the complex geometry of the moduli space; in particular, we compute dimensions, both from the complex geometric and the gauge theoretic point of view.

math.DG↗