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. Such a generalization is needed in order to use principal pairs as a general framework for describing augmented holomorphic bundles. We illustrate our results with applications to well known examples.
Let $X$ be a smooth projective complex variety with an ample line bundle $L$, and let $D$ be a simple normal crossing divisor. We establish the Kobayashi-Hitchin correspondence between tame harmonic bundles on $X-D$ and $μ_L$-stable parabolic $λ$-flat bundles with trivial characteristic numbers on $(X,D)$. Especially, we obtain the quasiprojective version of the Corlette-Simpson correspondence between flat bundles and Higgs bundles.
In the previous paper \cite{Goto_2017}, the notion of an Einstein-Hermitian metric of a generalized holomorphic vector bundle over a generalized Kahler manifold of symplectic type was introduced from the moment map framework. In this paper we establish a Kobayashi-Hitchin correspondence, that is, the equivalence of the existence of an Einstein-Hermitian metric and $ψ$-polystability of a generalized holomorphic vector bundle over a compact generalized Kahler manifold of symplectic type. Poisson modules provide intriguing generalized holomorphic vector bundles and we obtain $ψ$-stable Poisson mo…
We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure. In particular, our Hitchin-Kobayashi theorem holds on any compact Sasakian manifold. We define the notion of stability for foliated Hermitian vector bundles with transverse holomorphic structure and prove that such bundles admit a basic Hermitian-Einstein connection if and only if they are polystable. Our proof is obtained by adapting the proof by Uhlenbeck and Yau to the foliated setting. We relate the transverse Hermitian-Einstein equatio…
Recently, Donaldson proved asymptotic stability for a polarized algebraic manifold $M$ with polarization class admitting a Kähler metric of constant scalar curvature, essentially when the linear algebraic part $H$ of $Aut^0(M)$ is semisimple. The purpose of this paper is to give a generalization of Donaldson's result to the case where the polarization class admits an extremal Kähler metric, even when $H$ is not semisimple.
We prove the Kobayashi-Hitchin correspondence and the approximate Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if $X$ is a compact manifold and $g$ is a Gauduchon metric on $X$, a twisted holomorphic vector bundle on $X$ is $g-$polystable if and only if it is $g-$Hermite-Einstein, and if $X$ is a compact Kähler manifold and $g$ is a Kähler metric on $X$, then a twisted holomorphic vector bundle on $X$ is $g-$semistable if and only if it is approximate $g-$Hermite-Einstein.
We prove the existence of a Hermitian-Einstein metric on holomorphic vector bundles with a Hermitian metric satisfying the analytic stability condition, under some assumption for the underlying Kähler manifolds. We also study the curvature decay of the Hermitian-Einstein metrics. It is useful for the study of the classification of instantons and monopoles on the quotient of $4$-dimensional Euclidean space by some types of closed subgroups. We also explain examples of doubly periodic monopoles corresponding to some algebraic data.
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 …
In this paper, we introduce the notions of $α$-Hermitian-Einstein metric and $α$-stability for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Moreover, we establish a Kobayashi-Hitchin correspondence for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Examples of such vector bundles include generalized holomorphic bundles on generalized Kähler manifolds. We also show that $α$-stability of a vector bundle, in this sense, can depend on the parameter $α$.
In this paper, we establish the Hitchin--Kobayashi correspondence for the $I_\pm$-holomorphic quiver bundle $\mathcal{E}=(E,ϕ)$ over a compact generalized Kähler manifold $(X, I_+,I_-,g, b)$ such that $g$ is Gauduchon with respect to both $I_+$ and $I_-$, namely $\mathcal{E}$ is $(α,σ,τ)$-polystable if and only if $\mathcal{E}$ admits an $(α,σ,τ)$-Hermitian--Einstein metric.
We prove an analogue of the Kobayashi-Hitchin correspondence oncompact connected 3-folds that is fibered on orbifold Riemann surfaces and satisfy an integrability condition, which contains compact connected Sasakian 3-folds. We define mini-holomorphic bundles on such 3-folds and the algebraic Dirac-type singularities on mini-holomorphic bundles, and prove that there exists a special Hermitian metric (admissible BHE-metric) on a Dirac-type singular mini-holomorphic bundle if the bundle satisfies a slope stability.
A simple trick invoking objective B-fields is employed to refine the concept of characteristic classes for twisted bundles. Then the objective stability and objective Einstein metrics are introduced and a new Hitchin-Kobayashi correspondence is established between them. As an application the SO(3)-instanton moduli space is proved to be always orientable.
We establish the correspondence between tame harmonic bundles and $μ_L$-stable parabolic Higgs bundles with trivial characteristic numbers. We also show the Bogomolov-Gieseker type inequality for $μ_L$-stable parabolic Higgs bundles. Then we show that any local system on a smooth quasi projective variety can be deformed to a variation of polarized Hodge structure. As a consequence, we can conclude that some kind of discrete groups cannot be a split quotient of the fundamental group of a smooth quasi projective variety.
Using a quasi-linear version of Hodge theory, holomorphic vector bundles in a neighbourhood of a given polystable bundle on a compact Kaehler manifold are shown to be (poly)stable if and only if their corresponding classes are (poly)stable in the sense of geometric invariant theory with respect to the linear action of the automorphism group of the bundle on its space of infinitesimal deformations.
A twisted quiver bundle is a set of holomorphic vector bundles over a complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles, labelled by the arrows. When the manifold is Kaelher, quiver bundles admit natural gauge-theoretic equations, which unify many known equations for bundles with extra structure. In this paper we prove a Hitchin--Kobayashi correspondence for twisted quiver bundles over a compact Kaehler manifold, relating the existence of solutions to the gauge equations to a stability criterion, and…
T. Mochizuki constructs a theory of variations of wild Hodge structure for which the underlying flat connection can have irregular singularities at infinity. He extends in this way the correspondence of Corlette and Simpson between irreducible flat bundles and stables Higgs bundles, taking into account objects with irregular singularities. As an application, he proves a conjecture of Kashiwara concerning a generalization of the Hard Lefschetz theorem when the coefficients are the de Rham complex of a simple holonomic D-module on smooth complex projective variety.
We establish a Kobayashi-Hitchin correspondence for the stable Higgs sheaves on a compact Kaehler manifold. Using it, we also obtain a Kobayashi-Hitchin correspondence for the stable Higgs G-sheaves, where G is any complex reductive linear algebraic group.
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 compact Kaehler manifold and $E\to X$ a principal $K$ bundle, where $K$ is a compact connected Lie group. Let ${\cal A}^{1,1}$ be the set of connections on $E$ whose curvature lies in $Ω^{1,1}(E\times_{Ad} {\frak k})$, where ${\frak k}$ is the Lie algebra of $K$. Endow $\frak k$ with a nondegenerate biinvariant bilinear pairing. This allows to identify $\{\frak k}\simeq{\frak k}^*$. Let $F$ be a Kaehler left $K$-manifold and suppose that there exists a moment map $μ$ for the action of $K$ on $F$. Let ${\cal S}=Γ(E\times_K F)$. In this paper we study the equation $$ΛF_A+μ(Φ)=c$$ fo…
In this paper we develop a Kobayashi-Hitchin type correspondence between solutions of the extended Bogomolny equations on $Σ\times \RP$ with Nahm pole singularity at $Σ\times \{0\}$ and the Hitchin component of the stable $SL(2,\mathbb{R})$ Higgs bundle; this verifies a conjecture of Gaiotto and Witten. We also develop a partial Kobayashi-Hitchin correspondence for solutions with a knot singularity in this program, corresponding to the non-Hitchin components in the moduli space of stable $SL(2,\mathbb{R})$ Higgs bundles. We also prove existence and uniqueness of solutions with knot singulariti…
We prove a Hitchin-Kobayashi correspondence for extensions of Higgs bundles. The results generalize known results for extensions of holomorphic bundles. Using Simpson's methods, we construct moduli spaces of stable objects. In an appendix we construct Bott-Chern forms for Higgs bundles
We prove the Kobayashi-Hitchin correspondence between good wild harmonic bundles and polystable good filtered $λ$-flat bundles satisfying a vanishing condition. We also study the correspondence for good wild harmonic bundles with the homogeneity with respect to a group action, which is expected to provide another way to construct Frobenius manifolds.
The Hitchin-Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Luebke, Uhlenbeck and Yau, states that an indecomposable holomorphic vector bundle over a compact Kaehler manifold is stable in the sense of Takemoto-Mumford if and only if the vector bundle admits a Hermitian-Einstein metric. Its manifold analogue known as Yau's conjecture, which originated from Calabi's conjecture, asks whether ``stability'' and ``existence of extremal metrics'' for polarized manifolds are equivalent. In this note, the recent progress of this subject, by Donaldson, Tian and our grou…
We define a functional ${\cal J}(h)$ for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles \cite{Kobayashi}, and study some of its basic properties. We show that ${\cal J}(h)$ is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite-Yang-Mills metrics. We derive a formula relating ${\cal J}(h)$ and another functional ${\cal I}(h…
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 use the Kobayashi-Hitchin correspondence for parabolic bundles to reprove the results of Troyanov and Luo-Tian regarding existence and uniqueness of conformal spherical metrics on the Riemann sphere with prescribed cone angles in the interval $(0, 2π)$ at a given configuration of three or more points.
We introduce a vector bundle version of the complex Monge-Ampere equation motivated by a desire to study stability conditions involving higher Chern forms. We then restrict ourselves to complex surfaces, provide a moment map interpretation of it, and define a positivity condition (MA positivity) which is necessary for the infinite-dimensional symplectic form to be Kahler. On rank-2 bundles on compact complex surfaces, we prove two consequences of the existence of a "positively curved" solution to this equation - Stability (involving the second Chern character) and a Kobayashi-Lubke-Bogomolov-M…
We consider a version of Hermitian-Einstein equation but perturbed by a Higgs field with a solution called a Donaldson-Thomas instanton on compact Kähler threefolds. The equation could be thought of as a generalization of the Hitchin equation on Riemann surfaces to Kähler threefolds. In the appendix of arXiv:0805.2192, following an analogy with the Hitchin equation, we introduced a stability condition for a pair consisting of a locally-free sheaf over a compact Kähler threefold and a section of the associated sheaf of the endomorphisms tensored by the canonical bundle of the threefold. In this…
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…
The so-called Hitchin-Kobayashi correspondence, proved by Donaldson, Uhlenbeck and Yau, establishes that an indecomposable holomorphic vector bundle over a compact Kahler manifold admits a Hermitian-Einstein metric if and only if the bundle satisfies the Mumford-Takemoto stability condition. In this paper we consider a variant of this correspondence for G-equivariant vector bundles on the product of a compact Kahler manifold X by a flag manifold G/P, where G is a complex semisimple Lie group and P is a parabolic subgroup. The modification that we consider is determined by a filtration of the v…
We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.
We define Hitchin's moduli space for a principal bundle $P$, whose structure group is a compact semisimple Lie group $K$, over a compact non-orientable Riemannian manifold $M$. We use the Donaldson-Corlette correspondence, which identifies Hitchin's moduli space with the moduli space of flat $K^\mathbb{C}$-connections, which remains valid when M is non-orientable. This enables us to study Hitchin's moduli space both by gauge theoretical methods and algebraically by using representation varieties. If the orientable double cover $\tilde{M}$ of $M$ is a Kähler manifold with odd complex dimension …
We establish a Kobayashi-Hitchin correspondence between solutions of the extended Bogomolny equation with a Dirac type singularity and Hecke modifications of Higgs bundles. This correspondence was conjectured by Witten and plays an important role in the physical description of the the geometric Langlands program in terms of S-duality for N=4 super Yang-Mills theory in four dimensions.
We obtain a Hitchin-Kobayashi-type correspondence for symplectic vortex equations, with the target a Kahler cone over a compact Sasakian manifold. We show that the correspondence reduces to studying the existence and uniqueness of Kazdan-Warner equations. Using this, we construct a map between the moduli space of solutions to the symplectic vortex equations and effective divisors.
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.
In general, a Kobayashi-Hitchin correspondence establishes an isomorphism between a moduli space of stable algebraic geometric objects and a moduli space of solutions of a certain (generalized) Hermite-Einstein equation. We believe that, for a large class of moduli problems, this correspondence respects the virtual fundamental classes defined in the two categories. We prove this statement in an interesting case, namely the moduli problem associated with the symplectic factorization problem which yields the complete toric varieties. Therefore, our main objects are the moduli spaces of "twisted"…
The notion of flat $λ$-connections as the interpolation of usual flat connections and Higgs fields was suggested by Deligne and further studied by Simpson. Mochizuki established the Kobayashi--Hitchin-type theorem for $λ$-flat bundles ($λ\neq 0$), which is called the Mochizuki correspondence. In this paper, on the one hand, we generalize Mochizuki's result to the case when the base being a compact balanced manifold, more precisely, we prove the existence of harmonic metrics on stable $λ$-flat bundles ($λ\neq 0$). On the other hand, we study two applications of the Simpson--Mochizuki correspond…
This article describes a Hitchin-Kobayashi style correspondence for the Vafa-Witten equations on smooth projective surfaces. This is an equivalence between a suitable notion of stability for a pair $(\mathcal{E}, φ)$, where $\mathcal{E}$ is a locally-free sheaf over a surface $X$ and $φ$ is a section of $\text{End} (\mathcal{E}) \otimes K_{X}$; and the existence of a solution to certain gauge-theoretic equations, the Vafa-Witten equations, for a Hermitian metric on $\mathcal{E}$. It turns out to be a special case of results obtained by Alvarez-Consul and Garcia-Prada. In this article, we give …
We study a certain type of wild harmonic bundles in relation with a Toda equation. We explain how to obtain a classification of the real valued solutions of the Toda equation in terms of their parabolic weights, from the viewpoint of the Kobayashi-Hitchin correspondence. Then, we study the associated integrable variation of twistor structure. In particular, we give a criterion for the existence of an integral structure. It follows from two results. One is the explicit computation of the Stokes factors of a certain meromorphic flat bundle. The other is an explicit description of the associated …
Let $P$ be a parabolic subgroup of a connected simply connected complex semisimple Lie group $G$. Given a compact Kähler manifold $X$, the dimensional reduction of $G$-equivariant holomorphic vector bundles over $X\times G/P$ was carried out by the first and third authors. This raises the question of dimensional reduction of holomorphic principal bundles over $X\times G/P$. The method used for equivariant vector bundles does not generalize to principal bundles. In this paper, we adapt to equivariant principal bundles the Tannakian approach of Nori, to describe the dimensional reduction of $G$-…
In this paper, we prove \emph{a priori} estimates for some vortex-type equations on compact Riemann surfaces. As applications, we recover existing estimates for the vortex bundle Monge-Ampère equation, prove an existence and uniqueness theorem for the Calabi-Yang-Mills equations on vortex bundles, and get estimates for $J-$vortex equation. We prove an existence and uniqueness result relating Gieseker stability and the existence of almost Hermitian Einstein metrics, i.e., a Kobayashi-Hitchin type correspondence. We also prove Kählerness of the negative of the symplectic form which arises in the…
In this paper we study gauge theory on SL(2,C)-equivariant bundles over XxP^1, where X is a compact Kahler manifold, P^1 is the complex projective line, and the action of SL(2,C) is trivial on X and standard on P^1. We first classify these bundles, showing that they are in correspondence with objects on X - that we call holomorphic chains - consisting of a finite number of holomorphic bundles E_i and morphisms E_i->E_{i-1}. We then prove a Hitchin-Kobayashi correspondence relating the existence of solutions to certain natural gauge-theoretic equations and an appropriate notion of stability for…
We show that the extended principal bundle of a Cartan geometry of type $(A(m,\mathbb{R}),GL(m,\mathbb{R}))$, endowed with its extended connection $\hatω$, is isomorphic to the principal $A(m,\mathbb{R})$-bundle of affine frames endowed with the affine connection as defined in classical Kobayashi-Nomizu volume I. Then we classify the local holonomy groups of the Cartan geometry canonically associated to a Riemannian manifold. It follows that if the holonomy group of the Cartan geometry canonically associated to a Riemannian manifold is compact then the Riemannian manifold is locally a product …
We introduce real structures on $L$-twisted Higgs pairs over a compact Riemann surface equipped with an anti-holomorphic involution, and prove a Hitchin--Kobayashi correspondence for them. Real $G$-Higgs bundles, where $G$ is a real form of a connected semisimple complex affine algebraic group $G^{\mathbb{C}}$, constitute a particular class of examples of these pairs. The real structure in this case involves a conjugation of $G^{\mathbb{C}}$ commuting with the one defining the real form $G$. We establish a homeomorphism between the moduli space of real $G$-Higgs bundles and the moduli space of…
We develop a Kobayashi-Hitchin correspondence for the extended Bogomolny equations, i.e., the dimensionally reduced Kapustin-Witten equations, on the product of a compact Riemann surface $Σ$ with ${\mathbb R}^+_y$, with generalized Nahm pole boundary conditions at $y=0$. The correspondence is between solutions of these equations satisfying these singular boundary conditions and also limiting to flat connections as $y \to \infty$, and certain holomorphic data consisting of effective triplets $(\mathcal E, φ, L)$ where $(\mathcal E, φ)$ is a stable $\mathrm{SL}(n+1,\mathbb C)$ Higgs pair and $L …
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…
