Kobayashi–Hitchin correspondence

In differential geometry, algebraic geometry, and gauge theory, the Kobayashi–Hitchin correspondence (or Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. 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.[6][7]

The theorem can be considered a vast generalisation of the Narasimhan–Seshadri theorem concerned with the case of compact Riemann surfaces, and has been influential in the development of differential geometry, algebraic geometry, and gauge theory since the 1980s. In particular the Hitchin–Kobayashi correspondence inspired conjectures leading to the nonabelian Hodge correspondence for Higgs bundles, as well as the Yau–Tian–Donaldson conjecture about the existence of Kähler–Einstein metrics on Fano varieties, and the Thomas–Yau conjecture about existence of special Lagrangians inside isotopy classes of Lagrangian submanifolds of a Calabi–Yau manifold.[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. This conjecture was first shared explicitly by Kobayashi and Hitchin independently in the early 1980s.[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.[10][11] In that setting, a Hermitian Yang–Mills connection could be simply understood as a (projectively) flat connection over the Riemann surface. 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.[4] Due to their work, the Kobayashi–Hitchin correspondence is often also referred to as the Donaldson–Uhlenbeck–Yau theorem. 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.[14]

In the later 1980s, attention turned to establishing the correspondence not just in the case of compact Kähler manifolds, but also for arbitrary compact complex manifolds. There is difficulty in this setting in even defining the notion of stability. For non-Kähler manifolds one must use a Gauduchon metric to define stability, but this is no restriction as every metric on a compact complex manifold is conformal to a Gauduchon metric. In 1987 existence on arbitrary compact complex surfaces was shown by Buchdahl, and shortly after for arbitrary compact complex manifolds by Li–Yau.[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.[15][16][17]

The notion of stability was introduced in algebraic geometry by Mumford in his work on geometric invariant theory, with a view to constructing moduli spaces of various geometric objects.[18] Mumford applied this new theory vector bundles to develop a notion of slope stability.[19]

A vector bundle is slope polystable if it is isomorphic to a direct sum of stable holomorphic vector bundles of the same slope. A vector bundle is slope unstable if it is not slope semistable.

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.

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. Kobayashi and Lübke provided proofs of this direction.[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 Kobayashi–Hitchin correspondence was one of the first instances of a general principle that has come to dominate geometry research since its proof: . Many results have been proven either as extensions or variations of the Kobayashi–Hitchin correspondence, or by direct analogy with the correspondence to seemingly disparate parts of geometry, and all of these results follow along this same principle. Here a summary of these generalisations or related results is given:

extremal objects in differential geometry correspond to stable objects in algebraic geometry

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.[27][28]

The Kobayashi–Hitchin correspondence has found a variety of important applications throughout algebraic geometry, differential geometry, and differential topology. 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.[38] In particular, the moduli space of Hermite–Einstein vector bundles comes naturally equipped with a Riemannian structure, given by a Weil–Peterson-type metric on the moduli space. Combining this geometric structure with the natural algebraic compactifications of the moduli space arising out of the Kobayashi–Hitchin correspondence, given by the moduli spaces of slope semistable or Gieseker semistable sheaves, allows one to integrate characteristic classes over the moduli space to obtain invariants of the original complex manifold. This is most famously used in Donaldson theory, where invariants of smooth four-manifolds are obtained. Similar techniques have been used in Seiberg–Witten 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. This is one sense in which the correspondence has had lasting impacts in enumerative geometry.[39]