The correspondence is named after Shoshichi Kobayashi and Nigel Hitchin, who independently conjectured in the 1980s that the moduli spaces of stable vector bundles and Einstein–Hermitian vector bundles over a complex manifold were essentially the same.
[1][2] This was proven by Simon Donaldson for projective algebraic surfaces and later for projective algebraic manifolds,[3][4] by Karen Uhlenbeck and Shing-Tung Yau for compact Kähler manifolds,[5] and independently by Buchdahl for non-Kahler compact surfaces, and by Jun Li and Yau for arbitrary compact complex manifolds.
[8] In 1965, M. S. Narasimhan and C. S. Seshadri proved the Narasimhan–Seshadri theorem, which relates stable holomorphic (or algebraic) vector bundles over compact Riemann surfaces (or non-singular projective algebraic curves), to projective unitary representations of the fundamental group of the Riemann surface.
[9] It was realised in the 1970s by Michael Atiyah, Raoul Bott, Hitchin and others that such representation theory of the fundamental group could be understood in terms of Yang–Mills connections, notions arising out of then-contemporary mathematical physics.
Inspired by the Narasimhan–Seshadri theorem, around this time a folklore conjecture formed that slope polystable vector bundles admit Hermitian Yang–Mills connections.
This is partially due to the argument of Fedor Bogomolov and the success of Yau's work on constructing global geometric structures in Kähler geometry.
[1][2] The explicit relationship between Yang–Mills connections and stable vector bundles was made concrete in the early 1980s.
A direct correspondence when the dimension of the base complex manifold is one was explained in the work of Atiyah and Bott in 1982 on the Yang–Mills equations over compact Riemann surfaces, and in Donaldson's new proof of the Narasimhan–Seshadri theorem from the perspective of gauge theory in 1983.
The notion of a Hermitian–Einstein connection for a vector bundle over a higher dimensional complex manifold was distilled by Kobayashi in 1980, and in 1982 he showed in general that a vector bundle admitting such a connection was slope stable in the sense of Mumford.
[12][13] The more difficult direction of proving the existence of Hermite–Einstein metrics on stable holomorphic vector bundles over complex manifolds of dimension larger than one quickly followed in the 1980s.
Soon after providing a new proof of the Narasimhan–Seshadri theorem in complex dimension one, Donaldson proved existence for algebraic surfaces in 1985.
[3] The following year Uhlenbeck–Yau proved existence for arbitrary compact Kähler manifolds using a continuity method.
[5] Shortly after that Donaldson provided a second proof tailored specifically to the case of projective algebraic manifolds using the theory of determinant bundles and the Quillen metric.
In 2019 Karen Uhlenbeck was awarded the Abel prize in part for her work on the existence of Hermite–Einstein metrics, as well as her contributions to the key analytical techniques that underpin the proof of the theorem.
[6][7] The Kobayashi–Hitchin correspondence concerns the existence of Hermitian Yang–Mills connections (or Hermite–Einstein metrics) on holomorphic vector bundles over compact complex manifolds.
In this section the precise notions will be presented for the setting of compact Kähler manifolds.
[18] Mumford applied this new theory vector bundles to develop a notion of slope stability.
Namely in the algebraic setting the rank and degree of a coherent sheaf are encoded in the coefficients of its Hilbert polynomial, and the expressions for these quantities may be extended in a straightforward way to the setting of Kähler manifolds that aren't projective by replacing the ample line bundle by the Kähler class and intersection pairings by integrals.
It is possible to phrase the definition in terms of either the Hermitian metric itself, or its associated Chern connection, and the two notions are essentially equivalent up to gauge transformation.
Here we give the statement of the Kobayashi–Hitchin correspondence for arbitrary compact complex manifolds, a case where the above definitions of stability and special metrics can be readily extended.
The Kobayashi–Hitchin correspondence does not just imply a bijection of sets of slope polystable vector bundles and Hermite–Einstein metrics, but an isomorphism of moduli spaces.
Namely, two polystable holomorphic vector bundles are biholomorphic if and only if there exists a gauge transformation taking the corresponding Hermite–Einstein metrics from one to the other, and the map
up to gauge transformation.One direction of the proof of the Kobayashi–Hitchin correspondence, the stability of a holomorphic vector bundle admitting a Hermite–Einstein metric, is a relatively straightforward application of the principle in Hermitian geometry that curvature decreases in holomorphic subbundles.
[12][20] The main difficulty in this direction is to show stability with respect to coherent subsheaves which are not locally free, and to do this Kobayashi proved a vanishing theorem for sections of Hermite–Einstein vector bundles.
The more complicated direction of showing the existence of a Hermite–Einstein metric on a slope polystable vector bundle requires sophisticated techniques from geometric analysis.
Uhlenbeck and Yau proved the general case of the correspondence by applying a continuity method and showing that the obstruction to the completion of this continuity method can be characterised precisely by an analytic coherent subsheaf with which slope-destabilises the vector bundle.
Here a summary of these generalisations or related results is given: In addition to admitting many direct or vast generalisations, the Kobayashi–Hitchin correspondence has also served as a guiding result for other correspondences which do not directly fit into the framework of Hermitian metrics on vector bundles.
By providing two alternative descriptions of the moduli space of stable holomorphic vector bundles over a complex manifold, one algebraic in nature and the other analytic, many important results about such moduli spaces have been able to be proved.
The most spectacular of these has been to the study of invariants of four-manifolds and more generally to algebraic varieties, through Donaldson–Thomas theory.
In higher dimensions, Donaldson–Thomas theory and integration over virtual fundamental classes was developed in analogy with the dual descriptions of moduli spaces of sheaves that is afforded by the Kobayashi–Hitchin correspondence.