Nonabelian Hodge correspondence
In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence (named after Kevin Corlette and Carlos Simpson) 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 (in fact the terminology "Higgs bundle" was introduced by Carlos Simpson after the work of Hitchin). 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 formula_1 is a pair formula_2 where formula_3 is a holomorphic vector bundle and formula_4 is an formula_5-valued holomorphic formula_6-form on formula_7, called the Higgs field. Additionally, the Higgs field must satisfy formula_8. A Higgs bundle is (semi-)stable if, for every proper, non-zero coherent subsheaf  formula_9 which is preserved by the Higgs field, so that formula_10, one has This rational number is called the slope, denoted formula_12, 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 formula_2. A Hermitian metric formula_14  on a Higgs bundle formula_2 gives rise to a Chern connection formula_16 and curvature formula_17. The condition that formula_18 is holomorphic can be phrased as formula_19. Hitchin's equations, on a compact Riemann surface, state that for a constant formula_21. In higher dimensions these equations generalise as follows. Define a connection formula_22 on formula_23 by formula_24. This connection is said to be a Hermitian Yang–Mills connection (and the metric a Hermitian Yang–Mills metric if This reduces to Hitchin's equations for a compact Riemann surface. Representations of the fundamental group and harmonic metrics. A representation of the fundamental  group formula_26 gives rise to a vector bundle with flat connection as follows. The universal cover formula_27 of formula_7 is a principal bundle over formula_7 with structure group formula_30. Thus there is an associated bundle to formula_27 given by This vector bundle comes naturally equipped with a flat connection formula_22. If formula_14 is a Hermitian metric on formula_23, define an operator formula_36 as follows. Decompose formula_37 into operators of type formula_6 and formula_39, respectively. Let formula_40 be the unique operator of type formula_6 such that the formula_6-connection formula_43 preserves the metric formula_14. Define formula_45, and set formula_46. Define the  pseudocurvature of formula_14 to be formula_48. The metric formula_14 is said to be harmonic if Notice that the condition formula_51 is equivalent to the three conditions formula_52, so if formula_51 then the pair formula_2 defines a Higgs bundle with holomorphic structure on formula_23 given by the Dolbeault operator formula_56. It is a result of Corlette that if formula_14 is harmonic, then it automatically satisfies formula_51 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 formula_23. Every Higgs bundle will be considered to have the underlying smooth vector bundle formula_23. 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 formula_7, 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 together, 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 formula_93 are naturally complex algebraic varieties, and where it is smooth, the de Rham moduli space formula_94 is a Riemannian manifold. On the common locus where these moduli spaces are smooth, the map formula_95 is a diffeomorphism, and since formula_96 is a complex manifold on the smooth locus, formula_94 obtains a compatible Riemannian and complex structure, and is  therefore a Kähler manifold. Similarly, on the smooth locus, the map formula_98 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 formula_99 (for the associated integrable almost complex structures) then formula_100. In particular if one defines a third almost complex structure by formula_101 then formula_102. If one combines these three complex structures with the Riemannian metric coming from formula_94, 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 formula_18 to zero, then a Higgs bundle is simply a holomorphic vector bundle. This gives an inclusion formula_68 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, formula_106 . The subset of the Betti moduli space corresponding to the unitary representations, denoted formula_107, will get mapped isomorphically onto the moduli space of semi-stable vector bundles formula_108. Examples. Rank one Higgs bundles on compact Riemann surfaces. The special case where the rank of the underlying vector bundle is one gives rise to a simpler correspondence. Firstly, every line bundle is stable, as there are no proper non-zero subsheaves. In this case, a Higgs bundle consists of a pair formula_109 of a holomorphic line bundle and a holomorphic formula_6-form, since the endomorphism of a line bundle are trivial . In particular, the Higgs field is uncoupled from the holomorphic line bundle, so the moduli space formula_111 will split as a product, and the one-form automatically satisfies the condition formula_8. The gauge group of a line bundle is commutative, and so acts trivially on the Higgs field formula_18 by conjugation. Thus the moduli space can be identified as a product of the Jacobian variety of formula_7, classifying all holomorphic line bundles up to isomorphism, and the vector space formula_116 of holomorphic formula_6-forms. In the case of rank one Higgs bundles on compact Riemann surfaces, one obtains a further  description of the moduli space. The fundamental group of a compact Riemann surface, a surface group, is given by where formula_119 is the genus of the Riemann surface. The representations of formula_30 into the general linear group formula_121 are therefore given by formula_122-tuples of non-zero complex numbers: Since formula_124 is abelian, the conjugation on this space is trivial, and the Betti moduli space is formula_125. On the other hand, by Serre duality, the space of holomorphic formula_6-forms is dual to the sheaf cohomology formula_127. The Jacobian variety is an Abelian variety given by the quotient so has  tangent spaces given by the vector space formula_129, and cotangent bundle That is, the Dolbeault moduli space, the moduli space of holomorphic Higgs line bundles, is simply the cotangent bundle to the Jacobian, the moduli space of holomorphic line bundles. The nonabelian Hodge correspondence therefore gives a diffeomorphism which is not a biholomorphism. One can check that the natural complex structures on these two spaces are different, and satisfy the relation formula_100, giving a hyperkähler structure on the cotangent bundle to the Jacobian. Generalizations. It is possible to define the notion of a principal formula_133-Higgs bundle for a complex  reductive algebraic group formula_133, a version of Higgs bundles in the category of principal bundles. There is a notion of a stable principal bundle, and one can define a stable principal formula_133-Higgs bundle. A version of the nonabelian Hodge theorem holds for these objects, relating principal formula_133-Higgs bundles to representations of the fundamental group into formula_133. Nonabelian Hodge theory. The correspondence between Higgs bundles and representations of the fundamental group can be phrased as a kind of "nonabelian" Hodge theorem, which is to say, an analogy of the Hodge decomposition of a Kähler manifold, but with coefficients in  the nonabelian group formula_138 instead of the abelian group formula_139. The exposition here follows the discussion by Oscar Garcia-Prada in the appendix to Wells' "Differential Analysis on Complex Manifolds". Hodge decomposition. The Hodge decomposition of a compact Kähler manifold decomposes the complex de Rham cohomology into the finer Dolbeault cohomology: At degree one this gives a direct sum where we have applied the Dolbeault theorem to phrase the Dolbeault cohomology in terms of sheaf cohomology of the sheaf of holomorphic formula_6-forms formula_143 and the structure sheaf formula_144 of holomorphic functions on formula_7. Nonabelian cohomology. When constructing sheaf cohomology , the coefficient sheaf formula_146 is always a sheaf of abelian groups. This is because for an abelian group, every subgroup is normal, so the quotient group of sheaf cocycles by sheaf coboundaries is always well-defined. When the sheaf formula_146 is not abelian, these quotients are not necessarily well-defined, and so sheaf cohomology theories do not exist, except in the following special cases: A key example of nonabelian cohomology occurs when the coefficient sheaf is formula_155, the sheaf of holomorphic functions into the complex general linear group. In this case it is a well-known fact from Čech cohomology  that the cohomology set is in one-to-one correspondence with the set of holomorphic vector bundles of rank formula_157 on formula_7, up to isomorphism. Notice that there is a distinguished holomorphic vector bundle of rank formula_157, the trivial vector bundle, so this is actually a "cohomology pointed set". In the special case formula_160 the general linear group is the abelian group formula_161 of non-zero complex numbers with respect to multiplication. In this case one obtains the "group" of holomorphic line bundles up to isomorphism, otherwise known as the Picard group. Nonabelian Hodge theorem. The first cohomology group formula_162  is isomorphic to the group of homomorphisms from the fundamental group formula_30 to formula_139. This can be understood, for example, by applying the Hurewicz theorem. Thus the regular Hodge decomposition mentioned above may be phrased as The nonabelian Hodge correspondence gives an analogy of this statement of the Hodge theorem for nonabelian cohomology, as follows. A Higgs bundle consists of a pair formula_2 where formula_23 is a holomorphic vector bundle, and formula_168 is a holomorphic, endomorphism-valued formula_6-form. The holomorphic vector bundle formula_23 may be identified with an element of formula_156 as mentioned above. Thus a Higgs bundle may  be thought of as an element of the direct product The nonabelian Hodge correspondence gives an isomorphism from the moduli space of formula_173-representations of the fundamental group formula_30 to the moduli space of Higgs bundles, which could therefore be written as an isomorphism This can be seen as an analogy of the regular Hodge decomposition above. The moduli space of representations formula_176 plays the role of the first cohomology of formula_7 with nonabelian coefficients, the cohomology set formula_178 plays the role of the space formula_129, and the group formula_180 plays the role of the holomorphic (1,0)-forms formula_116. The isomorphism  here is written formula_182, but this is not an actual isomorphism of sets, as the moduli space of Higgs bundles is not literally given by the direct sum above, as this is only an analogy. Hodge structure. The moduli space formula_183 of semi-stable Higgs bundles has a natural action of the multiplicative group formula_124, given by scaling the Higgs field: formula_185 for formula_186. For abelian cohomology, such a formula_124 action gives rise to a "Hodge structure", which is a generalisation of the Hodge decomposition of the cohomology of a compact Kähler manifold. One way of understanding the nonabelian Hodge  theorem is to use the formula_124 action on the moduli space formula_96 to obtain a Hodge filtration. This can lead to new topological invariants of the underlying manifold formula_7. For example, one obtains restrictions on which groups may appear as the fundamental groups of compact Kähler manifolds in this way.