Nonabelian Hodge correspondence

In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact Kähler manifold.
The theorem can be considered a vast generalisation of the Narasimhan–Seshadri theorem which defines a correspondence between stable vector bundles and unitary representations of the fundamental group of a compact Riemann surface. In fact the Narasimhan–Seshadri theorem may be obtained as a special case of the nonabelian Hodge correspondence by setting the Higgs field to zero.

History

It was proven by M. S. Narasimhan and C. S. Seshadri in 1965 that stable vector bundles on a compact Riemann surface correspond to irreducible projective unitary representations of the fundamental group. This theorem was phrased in a new light in the work of Simon Donaldson in 1983, who showed that stable vector bundles correspond to Yang–Mills connections, whose holonomy gives the representations of the fundamental group of Narasimhan and Seshadri. The Narasimhan–Seshadri theorem was generalised from the case of compact Riemann surfaces to the setting of compact Kähler manifolds by Donaldson in the case of algebraic surfaces, and in general by Karen Uhlenbeck and Shing-Tung Yau. This correspondence between stable vector bundles and Hermitian Yang–Mills connections is known as the Kobayashi–Hitchin correspondence.
The Narasimhan–Seshadri theorem concerns unitary representations of the fundamental group. Nigel Hitchin introduced a notion of a Higgs bundle as an algebraic object which should correspond to complex representations of the fundamental group. The first instance of the nonabelian Hodge theorem was proven by Hitchin, who considered the case of rank two Higgs bundles over a compact Riemann surface. Hitchin showed that a polystable Higgs bundle corresponds to a solution of Hitchin's equations, a system of differential equations obtained as a dimensional reduction of the Yang–Mills equations to dimension two. It was shown by Donaldson in this case that solutions to Hitchin's equations are in correspondence with representations of the fundamental group.
The results of Hitchin and Donaldson for Higgs bundles of rank two on a compact Riemann surface were vastly generalised by Carlos Simpson and Kevin Corlette. The statement that polystable Higgs bundles correspond to solutions of Hitchin's equations was proven by Simpson. The correspondence between solutions of Hitchin's equations and representations of the fundamental group was shown by Corlette.

Definitions

In this section we recall the objects of interest in the nonabelian Hodge theorem.

Higgs bundles

A Higgs bundle over a compact Kähler manifold is a pair where is a holomorphic vector bundle and is an -valued holomorphic -form on, called the Higgs field. Additionally, the Higgs field must satisfy.
A Higgs bundle is stable if, for every proper, non-zero coherent subsheaf which is preserved by the Higgs field, so that, one has
This rational number is called the slope, denoted, and the above definition mirrors that of a stable vector bundle. A Higgs bundle is polystable if it is a direct sum of stable Higgs bundles of the same slope, and is therefore semi-stable.

Hermitian Yang–Mills connections and Hitchin's equations

The generalisation of Hitchin's equation to higher dimension can be phrased as an analog of the Hermitian Yang–Mills equations for a certain connection constructed out of the pair. A Hermitian metric on a Higgs bundle gives rise to a Chern connection and curvature. The condition that is holomorphic can be phrased as. Hitchin's equations, on a compact Riemann surface, state that
for a constant.
In higher dimensions these equations generalise as follows. Define a connection on by. This connection is said to be a Hermitian Yang–Mills connection if
This reduces to Hitchin's equations for a compact Riemann surface. Note that the connection is not a Hermitian Yang–Mills connection in the usual sense, as it is not unitary, and the above condition is a non-unitary analogue of the normal HYM condition.

Representations of the fundamental group and harmonic metrics

A representation of the fundamental group gives rise to a vector bundle with flat connection as follows. The universal cover of is a principal bundle over with structure group. Thus there is an associated bundle to given by
This vector bundle comes naturally equipped with a flat connection. If is a Hermitian metric on, define an operator as follows. Decompose into operators of type and, respectively. Let be the unique operator of type such that the -connection preserves the metric. Define, and set. Define the pseudocurvature of to be.
The metric is said to be harmonic if
Notice that the condition is equivalent to the three conditions, so if then the pair defines a Higgs bundle with holomorphic structure on given by the Dolbeault operator.
It is a result of Corlette that if is harmonic, then it automatically satisfies and so gives rise to a Higgs bundle.

Moduli spaces

To each of the three concepts: Higgs bundles, flat connections, and representations of the fundamental group, one can define a moduli space. This requires a notion of isomorphism between these objects. In the following, fix a smooth complex vector bundle. Every Higgs bundle will be considered to have the underlying smooth vector bundle.

' The group of complex gauge transformations acts on the set of Higgs bundles by the formula. If and denote the subsets of semistable and stable Higgs bundles, respectively, then one obtains moduli spaces where these quotients are taken in the sense of geometric invariant theory, so orbits whose closures intersect are identified in the moduli space. These moduli spaces are called the Dolbeault moduli spaces. Notice that by setting, one obtains as subsets the moduli spaces of semi-stable and stable holomorphic vector bundles and. It is also true that if one defines the moduli space of polystable Higgs bundles then this space is isomorphic to the space of semi-stable Higgs bundles, as every gauge orbit of semi-stable Higgs bundles contains in its closure a unique orbit of polystable Higgs bundles.
' The group complex gauge transformations also acts on the set of flat connections on the smooth vector bundle. Define the moduli spaces where denotes the subset consisting of irreducible flat connections which do not split as a direct sum on some splitting of the smooth vector bundle. These moduli spaces are called the de Rham moduli spaces.
' The set of representations of the fundamental group of is acted on by the general linear group by conjugation of representations. Denote by the superscripts and the subsets consisting of semisimple representations and irreducible representations respectively. Then define moduli spaces of semisimple and irreducible representations, respectively. These quotients are taken in the sense of geometric invariant theory, where two orbits are identified if their closures intersect. These moduli spaces are called the Betti moduli spaces'''.
Statement

The nonabelian Hodge theorem can be split into two parts. The first part was proved by Donaldson in the case of rank two Higgs bundles over a compact Riemann surface, and in general by Corlette. In general the nonabelian Hodge theorem holds for a smooth complex projective variety, but some parts of the correspondence hold in more generality for compact Kähler manifolds.
The second part of the theorem was proven by Hitchin in the case of rank two Higgs bundles on a compact Riemann surface, and in general by Simpson.
Combined, the correspondence can be phrased as follows:

In terms of moduli spaces

The nonabelian Hodge correspondence not only gives a bijection of sets, but homeomorphisms of moduli spaces. Indeed, if two Higgs bundles are isomorphic, in the sense that they can be related by a gauge transformation and therefore correspond to the same point in the Dolbeault moduli space, then the associated representations will also be isomorphic, and give the same point in the Betti moduli space. In terms of the moduli spaces the nonabelian Hodge theorem can be phrased as follows.
In general these moduli spaces will be not just topological spaces, but have some additional structure. For example, the Dolbeault moduli space and Betti moduli space are naturally complex algebraic varieties, and where it is smooth, the de Rham moduli space is a Riemannian manifold. On the common locus where these moduli spaces are smooth, the map is a diffeomorphism, and since is a complex manifold on the smooth locus, obtains a compatible Riemannian and complex structure, and is therefore a Kähler manifold.
Similarly, on the smooth locus, the map is a diffeomorphism. However, even though the Dolbeault and Betti moduli spaces both have natural complex structures, these are not isomorphic. In fact, if they are denoted then. In particular if one defines a third almost complex structure by then. If one combines these three complex structures with the Riemannian metric coming from, then on the smooth locus the moduli spaces become a Hyperkähler manifold.

Relation to Hitchin–Kobayashi correspondence and unitary representations

If one sets the Higgs field to zero, then a Higgs bundle is simply a holomorphic vector bundle. This gives an inclusion of the moduli space of semi-stable holomorphic vector bundles into the moduli space of Higgs bundles. The Hitchin–Kobayashi correspondence gives a correspondence between holomorphic vector bundles and Hermitian Yang–Mills connections over compact Kähler manifolds, and can therefore be seen as a special case of the nonabelian Hodge correspondence.
When the underlying vector bundle is topologically trivial, the holonomy of a Hermitian Yang–Mills connection will give rise to a unitary representation of the fundamental group,. The subset of the Betti moduli space corresponding to the unitary representations, denoted, will get mapped isomorphically onto the moduli space of semi-stable vector bundles.