Kobayashi-Hitchin Correspondence Research Articles

Meromorphic parahoric Higgs torsors and filtered Stokes G-local systems on curves

In this paper, we consider the wild nonabelian Hodge correspondence for principal G-bundles on curves, where G is a connected complex reductive group. We establish the correspondence under a “very good” condition on the irregular type of the meromorphic G-connections introduced by Boalch, and thus confirm a conjecture in [9, §1.5]. We first give a version of Kobayashi–Hitchin correspondence, which induces a one-to-one correspondence between stable meromorphic parahoric Higgs torsors of degree zero (Dolbeault side) and stable meromorphic parahoric connections of degree zero (de Rham side). Then, by introducing a notion of stability condition on filtered Stokes G-local systems, we prove a one-to-one correspondence between stable meromorphic parahoric connections of degree zero (de Rham side) and stable filtered Stokes G-local systems of degree zero (Betti side). When G=GLn(C), the main result in this paper reduces to that in [4].

The Kobayashi–Hitchin Correspondence of Generalized Holomorphic Vector Bundles Over Generalized Kähler Manifolds of Symplectic Type

Abstract In this paper, we establish the Kobayashi–Hitchin correspondence, that is, the equivalence of the existence of an Einstein–Hermitian metric and $\psi $-polystability of a generalized holomorphic vector bundle over a compact generalized Kähler manifold of symplectic type. Poisson modules provide intriguing generalized holomorphic vector bundles, and we obtain $\psi $-stable Poisson modules over complex surfaces that are not stable in the ordinary sense.

Parabolic bundles and spherical metrics

We use the Kobayashi-Hitchin correspondence for parabolic bundles to reprove the results of Troyanov [Trans. Amer. Math. Soc. 324 (1991), pp. 793-821] and Luo-Tian [Proc. Amer. Math. Soc. 116 (1992), pp. 1119-1129] regarding existence and uniqueness of conformal spherical metrics on the Riemann sphere with prescribed cone angles in the interval ( 0 , 2 π ) (0, 2\pi ) at a given configuration of three or more points.

Good Wild Harmonic Bundles and Good Filtered 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.

Kobayashi—Hitchin correspondence for twisted vector bundles

Abstract 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.

The extended Bogomolny equations with generalized Nahm pole boundary conditions, II

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 $\Sigma$ 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 $(\mathcal E, \varphi, L)$ where $(\mathcal E, \varphi)$ is a stable $\mathrm{SL}(n+1,\mathbb C)$ Higgs pair and $L \subset \mathcal E$ is a holomorphic line bundle. This corroborates a prediction of Gaiotto and Witten, and is an extension of our earlier paper \cite{HeMazzeo2017} which treats only the $\mathrm{SL}(2,\mathbb R)$ case.

A vector bundle version of the Monge-Ampère equation

We introduce a vector bundle version of the complex Monge-Ampère 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 Kähler. 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-Lübke-Bogomolov-Miyaoka-Yau type inequality. Finally, we prove a Kobayashi-Hitchin correspondence for a dimensional reduction of the aforementioned equation.

The extended Bogomolny equations and generalized Nahm pole boundary condition

In this paper we develop a Kobayashi-Hitchin type correspondence between solutions of the extended Bogomolny equations on $\Sigma\times \RP$ with Nahm pole singularity at $\Sigma \times \{0\}$ 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 singularities on $\mathbb{C}\ti\RP$.

Kobayashi–Hitchin correspondence for analytically stable bundles

We prove the existence of an Hermitian–Einstein metric on holomorphic vector bundles with an 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 quotients of four-dimensional Euclidean space by some types of closed subgroups. We also explain examples of doubly periodic monopoles corresponding to some algebraic data.

Hecke modifications of Higgs bundles and the extended Bogomolny equation

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 (2015), p. 668 and plays an important role in the physical description of the geometric Langlands program in terms of S–duality for N=4 super Yang–Mills theory in four dimensions.

Kobayashi-Hitchin correspondence for tame harmonic bundles and an application

Kobayashi-Hitchin correspondence for tame harmonic bundles and an application

On the Donaldson–Uhlenbeck compactification of instanton moduli spaces on class VII surfaces

We study the following question: Let (X,g) be a compact Gauduchon surface, (E,h) be a differentiable rank r vector bundle on X⁠, D be a fixed holomorphic structure on D≔det(E) and a be the Chern connection of the pair (D,det(h))⁠. Does the complex space structure on MaASD(E)* induced by the Kobayashi–Hitchin correspondence extend to a complex space structure on the Donaldson–Uhlenbeck compactification M¯aASD(E)? Our results answer this question in detail for the moduli spaces of SU(2)-instantons with c2=1 on general (possibly unknown) class VII surfaces.

Convergence of the Yang–Mills–Higgs flow on Gauged Holomorphic maps and applications

The symplectic vortex equations admit a variational description as global minimum of the Yang–Mills–Higgs functional. We study its negative gradient flow on holomorphic pairs [Formula: see text] where [Formula: see text] is a connection on a principal [Formula: see text]-bundle [Formula: see text] over a closed Riemann surface [Formula: see text] and [Formula: see text] is an equivariant map into a Kähler Hamiltonian [Formula: see text]-manifold. The connection [Formula: see text] induces a holomorphic structure on the Kähler fibration [Formula: see text] and we require that [Formula: see text] descends to a holomorphic section of this fibration.We prove a Łojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the [Formula: see text]-topology when [Formula: see text] is equivariantly convex at infinity with proper moment map, [Formula: see text] is holomorphically aspherical and its Kähler metric is analytic.As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang–Mills–Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet’s Kobayashi–Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment–weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem.

A Kobayashi–Hitchin correspondence for I±-holomorphic bundles

In this paper, we introduce the notions of α-Hermitian–Einstein metric and α-stability for I±-holomorphic vector bundles on bi-Hermitian manifolds. Moreover, we establish a Kobayashi–Hitchin correspondence for I±-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.

Irreducible vector bundles on some elliptic non-Kahler threefolds

We study rank-2 vector bundles on non-Kähler threefolds π : X → B , which are elliptic principal bundles with at least one non-zero Chern class over a complex surface B with no curves. In this case, we prove that every rank-2 irreducible vector bundle on X is a pull-back from B up to a twist by a line bundle. These 2-vector bundles are, via the Kobayashi–Hitchin correspondence, solutions of the Yang–Mills equations on the threefold X .

Harmonic Bundles and Toda Lattices With Opposite Sign II

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 meromorphic flat bundle. We use the opposite filtration of the limit mixed twistor structure with an induced torus action.

Kähler–Einstein Metrics and Stability

This is a note to announce a solution of the well-known question: which Fano manifolds admit Kahler–Einstein metrics? The idea that the appropriate condition should be in terms of “algebro-geometric stability” was proposed about 20 years ago by Yau [21, 22] (partly by analogy with the “Kobayashi–Hitchin correspondence” in the case of holomorphic bundles). For more on the history of this problem please see our forthcoming full paper. Over the years various different notions of stability have been discussed in the literature, both in the Fano/Kahler–Einstein case and in the more general situation of constant scalar curvature Kahler metrics on polarized manifolds. But the condition we end up using is essentially as proposed by Tian [20] and which we now recall (see also [14] for the equivalence with a priori stronger definitions).

YANG–MILLS EQUATION FOR STABLE HIGGS SHEAVES

We establish a Kobayashi–Hitchin correspondence for the stable Higgs sheaves on a compact Kähler 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.

Affine Hermitian-Einstein Metrics

Here c1(E, h) is the first Chern form of E with respect to a Hermitian metric h. The famous theorem of Donaldson [7, 8] (for algebraic manifolds only) and Uhlenbeck-Yau [24, 25] says that an irreducible vector bundle E → N is ω-stable if and only if it admits a HermitianEinstein metric (i.e. a metric whose curvature, when the 2-form part is contracted with the metric on N , is a constant times the identity endomorphism on E). This correspondence between stable bundles and Hermitian-Einstein metrics is often called the Kobayashi-Hitchin correspondence. An important generalization of this theorem is provided by Li-Yau [15] for complex manifolds (and subsequently due to Buchdahl by a different method for surfaces [3]). The major insight for this extension is the fact that the degree is well-defined as long as the Hermitian form ω on N satisfies only ∂∂ωn−1 = 0. This is because

Kobayashi–Hitchin correspondence for tame harmonic bundles II

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 $\mu_L$-stable parabolic $\lambda$-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.