We provide a construction of the moduli spaces of framed Hitchin pairs and their master spaces. These objects have come to interest as algebraic versions of solutions of certain coupled vortex equations by work of Lin and Stupariu. Our method unifies and generalizes constructions of several similar moduli spaces. Here are some points which are also of interest in other similar situations: - Our construction does not require the symmetricity condition that the map ^2G x E -> E be zero, usually appearing in the context of Higgs bundles. - We carry out a detailed analysis of the polynomial stability parameter without referring to GIT. This sheds some light on intrinsic properties of such parameter dependent stability concepts. - The construction corrects an inaccuracy in our previous construction of the compactification of the Hitchin space and generalizes it.
We identify dglas that control infinitesimal deformations of the pairs (manifold, Higgs bundle) and of Hitchin pairs. As a consequence, we recover known descriptions of first order deformations and we refine known results on obstructions. Secondly we prove that the Hitchin map is induced by a natural L-infinity morphism and, by standard facts about L-infinity algebras, we obtain new conditions on obstructions to deform Hitchin pairs.
We define the moduli problem of Hitchin pairs over Deligne-Mumford Stack and prove this moduli problem is represented by a separated and locally finitely presented algebraic space, which is considered as the moduli space of Hitchin pairs over Deligne-Mumford stack.
Let $X$ be a compact connected Riemann surface and $D$ an effective divisor on $X$. Let ${\mathcal N}_H(r,d)$ denote the moduli space of $D$-twisted stable Higgs bundles (a special class of Hitchin pairs) on $X$ of rank $r$ and degree $d$. It is known that ${\mathcal N}_H(r,d)$ has a natural holomorphic Poisson structure which is in fact symplectic if and only if $D$ is the zero divisor. We prove that ${\mathcal N}_H(r,d)$ admits a natural enhancement to a holomorphic symplectic manifold which is called here ${\mathcal M}_H(r,d)$. This ${\mathcal M}_H(r,d)$ is constructed by trivializing, over…
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.
In this article, we study the deformation theory of locally free sheaves and Hitchin pairs over a nodal curve. As a special case, the infinitesimal deformation of these objects gives the tangent space of the corresponding moduli spaces, which can be used to calculate the dimension of the corresponding moduli space. We show that the deformation of locally free sheaves and Hitchin pairs over a nodal curve is equivalent to the deformation of generalized parabolic bundles and generalized parabolic Hitchin pairs over the normalization of the nodal curve respectively.
We construct a degeneration of the moduli space of Hitchin pairs on smooth projective curves when the curve degenerates to an irreducible curve with a single node. The degeneration constructed here is analogous to the models constructed by Gieseker and Nagaraj-Seshadri for the case of the usual moduli spaces (i.e when the Higgs structure is trivial). There is an canonical relative Hitchin map which is shown to be proper and the general fibre of the relative Hitchin map provides a new compactification of the Picard variety of smooth curves with normal crossing singularities.
Hitchin pairs on Riemann surfaces are generalizations of Higgs bundles, allowing the Higgs field to be twisted by an arbitrary line bundle. We consider this generalization in the context of $G$-Higgs bundles for a real reductive Lie group $G$. We outline the basic theory and review some selected results, including recent results by Nozad and the author arXiv:1602.02712 [math.AG] on Hitchin pairs for the unitary group of indefinite signature $\mathrm{U}(p,q)$.
Let $X$ be a compact Riemann surface $X$ of genus at--least two. Fix a holomorphic line bundle $L$ over $X$. Let $\mathcal M$ be the moduli space of Hitchin pairs $(E ,ϕ\in H^0(End(E)\otimes L))$ over $X$ of rank $r$ and fixed determinant of degree $d$. We prove that, for some numerical conditions, $\mathcal M$ is irreducible, and that the isomorphism class of the variety $\mathcal M$ uniquely determines the isomorphism class of the Riemann surface $X$.
This paper deals with moduli spaces of framed principal bundles with connections with irregular singularities over a compact Riemann surface. These spaces have been constructed by Boalch by means of an infinite-dimensional symplectic reduction. It is proved that the symplectic structure induced from the Atiyah--Bott form agrees with the one given in terms of hypercohomology. The main results of this paper adapt work of Krichever and of Hurtubise to give an interpretation of some Hitchin Hamiltonians as yielding Hamiltonian vector fields on moduli spaces of irregular connections that arise from…
We consider the problem of finding the limit at infinity (corresponding to the downward Morse flow) of a Higgs bundle in the nilpotent cone under the natural $\mathbb{C}^*$-action on the moduli space. For general rank we provide an answer for Higgs bundles with regular nilpotent Higgs field, while in rank three we give the complete answer. Our results show that the limit can be described in terms of data defined by the Higgs field, via a filtration of the underlying vector bundle.
We give an algebraic geometric compactification of certain moduli spaces of semistable E-pairs in the sense of Yokogawa. In particular, we obtain a compactification of the moduli spaces of semistable Higgs pairs on a curve which were constructed by N. Hitchin.
We give an algebraic approach to the study of Hitchin pairs and prove the tensor product theorem for Higgs semistable Hitchin pairs over smooth projective curves defined over algebraically closed fields $k$ of characteristic $0$ and characteristic $p$, with $p$ satisfying some natural bounds. We also prove the corresponding theorem for polystable bundles.
Fix integers $g\geq 3$ and $r\geq 2$, with $r\geq 3$ if $g=3$. Given a compact connected Riemann surface $X$ of genus $g$, let $\MDH(X)$ denote the corresponding $\text{SL}(r, {\mathbb C})$ Deligne--Hitchin moduli space. We prove that the complex analytic space $\MDH(X)$ determines (up to an isomorphism) the unordered pair $\{X, \overline{X}\}$, where $\overline{X}$ is the Riemann surface defined by the opposite almost complex structure on $X$.
The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category $\mathbf{SH}(k)$ in terms of Voevodsky's framed correspondences. In particular the motivically fibrant $Ω$-resolution in positive degrees of the motivic suspension spectrum $Σ_{\mathbb P^1}^\infty X_+$, where $X_+=X\amalg *$, for a smooth scheme $X\in \mathrm{Sm}_k$ over an infinite perfect field $k$, is computed. The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres $(\mathbb A^l\times X,(\mathbb A^l-0)\times X)$, $X\in \mathrm{Sm}_k$, is on…
A conjectural recursive relation for the Poincaré polynomial of the Hitchin moduli space is derived from wallcrossing in the refined local Donaldson-Thomas theory of a curve. A doubly refined generalization of this theory is also conjectured and shown to similarly determine the Hodge polynomial of the same moduli space.
A principal pair consists of a holomorphic principal $G$-bundle together with a holomorphic section of an associated Kaehler fibration. Such objects support natural gauge theoretic equations coming from a moment map condition, and also admit a notion of stability based on Geometric Invariant Theory. The Hitchin--Kobayashi correspondence for principal pairs identifies stability as the condition for the existence of solutions to the equations. In this paper we generalize these features in a way which allows the full gauge group of the principal bundle to be replaced by certain proper subgroups. …
Given a vector bundle $E$ on a smooth projective curve or surface $X$ carrying the structure of a $V$-twisted Hitchin pair for some vector bundle $V$, we observe that the associated tautological bundle $E^{[n]}$ on the punctual Hilbert scheme of points $X^{[n]}$ has an induced structure of a $((V^\vee)^{[n]})^\vee$-twisted Hitchin pair, where $(V^\vee)^{[n]}$ is a vector bundle on $X^{[n]}$ constructed using the dual $V^\vee$ of $V$. In particular, a Higgs bundle on $X$ induces a logarithmic Higgs bundle on the Hilbert scheme $X^{[n]}$. We then show that the known results on stability of tauto…
We develop a complete Hitchin-Kobayashi correspondence for twisted pairs on a compact Riemann surface X. The main novelty lies in a careful study of the the notion of polystability for pairs, required for having a bijective correspondence between solutions to the Hermite-Einstein equations, on one hand, and polystable pairs, on the other. Our results allow us to establish rigorously the homemomorphism between the moduli space of polystable G-Higgs bundles on X and the character variety for representations of the fundamental group of X in G. We also study in detail several interesting examples …
This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic bundles and generalised parabolic $L$-twisted Hitchin pairs on its normalisation as well as on an analogue of the Hitchin map for generalised parabolic $L$-twisted Hitchin pairs.
Let $X$ be a compact connected Riemann surface, $D\, \subset\, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x\, \subsetneq\, G_x$ a Zariski closed subgroup for every $x\, \in\, D$. A framed principal $G$--bundle is a pair $(E_G,\, ϕ)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $ϕ$ assigns to each $x\, \in\, D$ a point of the quotient space $(E_G)_x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,\, ϕ)$ together with a section $θ\, \in\, H^0(X,\, \text{ad}(E_G)\otimes K_X\otimes{\mathcal O}_X(D))$ su…
We show that there exists a fine moduli space for torsion-free sheaves on a projective surface, which have a "good framing" on a big and nef divisor. This moduli space is a quasi-projective scheme. This is accomplished by showing that such framed sheaves may be considered as stable pairs in the sense of Huybrechts and Lehn. We characterize the obstruction to the smoothness of the moduli space, and discuss some examples on rational surfaces.
We exhibit the elliptic Calogero-Moser system as a Hitchin system of G-principal Higgs pairs. The group G, though naturally associated to any root system, is not semi-simple. We then interpret the Lax pairs with spectral parameter of [dP1] and [BSC1] in terms of equivariant embeddings of the Hitchin system of G into that of GL(N).
Let X be an irreducible smooth complex projective curve of genus g>2, and let x be a fixed point. A framed bundle is a pair (E,ϕ), where E is a vector bundle over X, of rank r and degree d, and ϕ:E_x\to C^r is a non-zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter τ>0, which gives rise to the moduli space of τ-semistable framed bundles M^τ. We prove a Torelli theorem for M^τ, for τ>0 small enough, meaning, the isomorphism class of the one-pointed curve (X,x), and also the integer r, are uniquely determined by the isomorphism class of the varie…
We prove a very general Kobayashi-Hitchin correspondence on arbitrary compact Hermitian manifolds. This correspondence refers to moduli spaces of "universal holomorphic oriented pairs". Most of the classical moduli problems in complex geometry (e. g. holomorphic bundles with reductive structure groups, holomorphic pairs, holomorphic Higgs pairs, Witten triples, arbitrary quiver moduli problems) are special cases of this universal classification problem. Our Kobayashi-Hitchin correspondence relates the complex geometric concept "polystable oriented holomorphic pair" to the existence of a reduct…
Let X be a complex space and F a coherent O_X-module. A F-(co)framed} sheaf on X is a pair (E,f) with a coherent O_X-module E and a morphism of coherent sheaves f : F -> E (resp. f : E -> F). Two such pairs (E,f) and (E',f') are said to be isomorphic if there exists an isomorphism of sheaves g : E -> E' such that gof = f' (resp. f'og = f). A pair (E,f) is called simple if its only automorphism is the identity on E. In this note we prove a representability theorem in a relative framework, which implies in particular that there is a moduli space of simple F-(co)framed sheaves on a given compact …
Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve; we study the moduli problem of parametrizing certain pairs consisting of a sheaf E on S and a map of E to a fixed reference sheaf on S. We prove that the full moduli stack for this problem is representable by a scheme in some cases. Moreover, the moduli stack admits an action by the group ${\mathbf C}^*$, and we determine its fixed-point set, which leads in some special cases to explicit formulas for the rational homology of the moduli space.
Framed quiver moduli parametrize stable pairs consisting of a quiver representation and a map to a fixed graded vector space. Geometric properties and explicit realizations of framed quiver moduli for quivers without oriented cycles are derived, with emphasis on their cohomology. Their use for quantum group constructions is discussed.
Given a normal projective irreducible stack $\mathscr X$ over an algebraically closed field of characteristic zero we consider framed sheaves on $\mathscr X$, i.e., pairs $(\mathcal E,ϕ_{\mathcal E})$, where $\mathcal E$ is a coherent sheaf on $\mathscr X$ and $ϕ_{\mathcal E}$ is a morphism from $\mathcal E$ to a fixed coherent sheaf $\mathcal F$. After introducing a suitable notion of (semi)stability, we construct a projective scheme, which is a moduli space for semistable framed sheaves with fixed Hilbert polynomial, and an open subset of it, which is a fine moduli space for stable framed sh…
This article is an expanded version of talks given by the authors in Oberwolfach, Bochum, and at the Fano Conference in Torino. Some new results (e. g. the material concerning flag varieties, Quot spaces over $¶^1$, and the generalized quiver representations) were included. The main goal is the construction of gauge theoretical Gromov-Witten type invariants of arbitrary genus associated with certain symplectic factorization problems with additional symmetry, and the computation of these invariants in terms of complex geometric objects. The main tool for describing moduli spaces associated with…
A twisted Higgs bundle on a Kähler manifold $X$ is a pair $(E,ϕ)$ consisting of a holomorphic vector bundle $E$ and a holomorphic bundle morphism $ϕ\colon M\otimes E \to E$ for some holomorphic vector bundle $M$. Such objects were first considered by Hitchin when $X$ is a curve and $M$ is the tangent bundle of $X$, and also by Simpson for higher dimensional base. The Hitchin-Kobayashi correspondence for such pairs states that $(E,ϕ)$ is polystable if and only if $E$ admits a hermitian metric solving the Hitchin equation. This correspondence is a powerful tool to decide whether there exists a s…
We find an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL(2) Higgs moduli space on a Riemann surface. On one side we have the components of the Lagrangian brane of U(1,1) Higgs bundles whose mirror was proposed by Nigel Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL(2) Higgs moduli space. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchin's proposal.
In this paper the moduli space of Higgs pairs over a fixed smooth projective curve with extra formal data is defined and it is endowed with a scheme structure. We introduce a relative version of the Krichever map using a fibration of Sato Grassmannians and show that this map is injective. This fact and the characterization of the points of the image of the Krichever map allow us to prove that this moduli space is a closed subscheme of the product of the moduli of vector bundles (with formal extra data) and a formal anologue of the Hitchin base. This characterization also provide us the method …
A $\mathrm{U}(p,q)$-Higgs bundle on a Riemann surface (twisted by a line bundle) consists of a pair of holomorphic vector bundles, together with a pair of (twisted) maps between them. Their moduli spaces depend on a real parameter $α$. In this paper we study wall crossing for the moduli spaces of $α$-polystable twisted $\mathrm{U}(p,q)$-Higgs bundles. Our main result is that the moduli spaces are birational for a certain range of the parameter and we deduce irreducibility results using known results on Higgs bundles. Quiver bundles and the Hitchin-Kobayashi correspondence play an essential rol…
We describe stability conditions for pairs consisting of a coherent sheaf and a homomorphism to a fixed coherent sheaf on a projective variety. The corresponding moduli spaces are constructed for pairs on curves and surfaces. We consider two examples. The fixed sheaf is the structure sheaf or is a vector bundle on a divisor, i.e. Higgs pairs or framed bundles, resp. (unencoded version)
Let $X$ be a complex irreducible smooth projective curve, and let ${\mathbb L}$ be an algebraic line bundle on $X$ with a nonzero section $σ_0$. Let $\mathcal{M}$ denote the moduli space of stable Hitchin pairs $(E,\, θ)$, where $E$ is an algebraic vector bundle on $X$ of fixed rank $r$ and degree $δ$, and $θ\, \in\, H^0(X,\, End(E)\otimes K_X\otimes{\mathbb L})$. Associating to every stable Hitchin pair its spectral data, an isomorphism of $\mathcal{M}$ with a moduli space $\mathcal{P}$ of stable sheaves of pure dimension one on the total space of $K_X\otimes{\mathbb L}$ is obtained. Both the…
The Betti numbers of moduli spaces of representations of a universal central extension of a surface group in the groups U(2,1) and SU(2,1) are calculated. The results are obtained using the identification of these moduli spaces with moduli spaces of Higgs bundles, and Morse theory, following Hitchin's programme. This requires a careful analysis of critical submanifolds which turn out to have a description using either symmetric products of the surface or moduli spaces of Bradlow pairs.
We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such data, we associate an explicit meromorphic Higgs bundle on a genus-$g$ Riemann surface, where $g$ is the number of loops in the comet, thereby embedding the Nakajima quiver variety into a Hitchin system on a punctured genus-$g$ Riemann surface (generally with positive codimension).…
We introduce the concept of Spin^G-structure in a SO-bundle, where $G\subset U(V)$ is a compact Lie group containing $-id_V$. We study and classify $Spin^G(4)$-structures on 4-manifolds, we introduce the G-Monopole equations associated with a $Spin^G$-structure. On Kaehler surfaces a Kobayashi-Hitchin correspondence can be proved for the corresponding moduli spaces. Using this complex geometric interpretation, we determine explicitely a moduli space of "PU(2)-Monopoles" on $¶^2$, we describe its Uhlenbeck compactification, as well as the Donaldson- and the abelian locus.
We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of $G$-bundles on a smooth projective curve for a reductive algebraic group $G$. For example, our result applies to the stack of semistable $G$-bundles, stacks of semistable Hitchin pairs, and stacks of semistable parabolic $G$-bundles. Similar arguments apply to Gieseker semistable $G$-bundles in higher dimensions. We present two applications of the main result. First, we show that in characteristic $0$ every stack of semistable de…
The ADHM construction establishes a one-to-one correspondence between framed torsion free sheaves on the projective plane and stable framed representations of a quiver with relations in the category of complex vector spaces. This paper studies the geometry of moduli spaces of representations of the same quiver with relations in the abelian category of coherent sheaves on a smooth complex projective curve $X$. In particular it is proven that this moduli space is virtually smooth and related byrelative Beilinson spectral sequence to the curve counting construction via stable pairs of Pandharipan…
We give a natural construction of unramified over Z framed mixed Tate motives, whose periods are the multiple zeta values. Namely, for each convergent multiple zeta-value we define two boundary divisors A and B in the moduli space M_{0,n+3} of stable curves of genus zero. The corresponding multiple zeta-motive is the n-th cohomology of the pair (M_{0,n+3} -A,B).
We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type $\operatorname{SL}_n$ and $\operatorname{PGL}_n$. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for $d$ coprime to $n$, that the number of rank $n$ Higgs bundles of deg…
A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite torsion. In the language of Hitchin's and Gualtieri's generalized complex geometry, (4,4)-ma…
This expository paper details the theory of rank one Higgs bundles over a closed Riemann surface X and their relationship to representations of the fundamental group of X. We construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. The moduli spaces are real Lie groups. From each context arises a complex structure, and the different complex structures define a hyper-Kaehlerstructure. The twistor space, real forms, and various group actions are computed explicitly in terms of th…
In this paper, we give a natural construction of mixed Tate motives whose periods are a class of iterated integrals which include the multiple polylogarithm functions. Given such an iterated integral, we construct two divisors $A$ and $B$ in the moduli spaces $\mathcal{\bar{M}}_{0,n}$ of $n$-pointed stable curves of genus 0, and prove that the cohomology of the pair $(\mathcal{\bar{M}}_{0,n}-A,B-B\cap A)$ is a framed mixed Tate motive whose period is that integral. It generalizes the results of A. Goncharov and Yu. Manin for multiple zeta values. Then we apply our construction to the dilogarit…
We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincaré pairing) in terms of representation theory of some vertex operator algebras ("$\mathscr W$-algebras").
