Kobayashi–Hitchin correspondence

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]

Define the degree of a holomorphic vector bundle ${\displaystyle E\to (X,\omega )}$ over a compact Kähler manifold to be the integer

${\displaystyle \mathrm {deg} (E):=(c_{1}(E)\cup [\omega ]^{n-1})[X]}$

where ${\displaystyle c_{1}(E)}$ is the first Chern class of ${\displaystyle E}$. The slope of ${\displaystyle E}$ is the rational number ${\displaystyle \mu (E)}$ defined by

${\displaystyle \mu (E):={\frac {\mathrm {deg} (E)}{\mathrm {rank} (E)}}.}$

It is possible to extend the definition of slope to any analytic coherent sheaf over ${\displaystyle (X,\omega )}$. 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.

A holomorphic vector bundle ${\displaystyle E\to (X,\omega )}$ is said to be slope stable (resp. slope semistable) if for all proper, non-zero coherent subsheaves ${\displaystyle {\mathcal {F}}\subset E}$ with ${\displaystyle 0<\operatorname {rk} ({\mathcal {F}})<\operatorname {rk} (E)}$, the following inequality is satisfied:

${\displaystyle \mu ({\mathcal {F}})<\mu (E)\quad {\text{(resp. }}\leq {\text{)}}.}$

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.

The notion of a Hermitian Yang–Mills connection is a specification of a Yang–Mills connection to the case of a Hermitian vector bundle over a complex manifold. 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. Given a Hermitian vector bundle ${\displaystyle (E,h)\to (X,\omega )}$ over a compact Kähler manifold, a Hermitian Yang–Mills connection is a unitary connection ${\displaystyle A}$ for the Hermitian metric ${\displaystyle h}$ which satisfies

${\displaystyle {\begin{cases}F_{A}^{0,2}=0\\\Lambda _{\omega }F_{A}=\lambda (E)\operatorname {Id} _{E}.\end{cases}}}$

The condition that ${\displaystyle F_{A}^{0,2}=0}$ implies that the differential operator ${\displaystyle \nabla _{A}^{0,1}}$ is a Dolbeault operator for a holomorphic structure on the Hermitian vector bundle ${\displaystyle (E,h)}$, and that ${\displaystyle A}$ itself is the Chern connection for this holomorphic structure. The constant ${\displaystyle \lambda (E)\in \mathbb {C} }$ depends only on the topology of ${\displaystyle E}$, and can be computed to be

${\displaystyle \lambda (E)=-{\frac {2\pi i}{(n-1)!\mathrm {vol} (X)}}\mu (E).}$

If one instead starts with a holomorphic vector bundle ${\displaystyle E\to (X,\omega )}$ and varies the choice of Hermitian metric, then a solution of the above equations, where ${\displaystyle A}$ is the Chern connection of the Hermitian metric, is called a Hermite–Einstein metric.

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.

Theorem (Donaldson–Uhlenbeck–Yau, Budhahl, Li–Yau): A holomorphic vector bundle ${\displaystyle E\to (X,\omega )}$ over a compact complex manifold with metric 2-form ${\displaystyle \omega }$ admits a Hermite–Einstein metric if and only if it is slope polystable.

If one instead restricts to irreducible holomorphic vector bundles, then slope polystability may be replaced with slope stability. 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 ${\displaystyle h\mapsto E}$ taking a Hermite–Einstein metric to its corresponding polystable vector bundle is continuous with respect to taking sequences of Hermitian metrics and holomorphic vector bundles in the appropriate topologies. Thus one may state the correspondence as follows:

Theorem (Moduli space version): There is a homeomorphism of the moduli space of polystable holomorphic vector bundles over ${\displaystyle (X,\omega )}$ with fixed underlying smooth structure ${\displaystyle E\to (X,\omega )}$ up to biholomorphism, and the moduli space of Hermite–Einstein metrics on the complex vector bundle ${\displaystyle E}$ 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. 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 more complicated direction of showing the existence of a Hermite–Einstein metric on a slope polystable vector bundle requires sophisticated techniques from geometric analysis. Many of these techniques build on the ideas developed by Yau in his proof of the Calabi conjecture, as well as on the important work of Uhlenbeck on harmonic maps in the 1970s, and her important analytical results about Yang–Mills connections from the early 1980s. 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. These techniques were built on by Buchdahl and Li–Yau in the setting where the 2-form ${\displaystyle \omega }$ is not closed, so that the compact complex manifold is not Kähler.[6][7]

• A form of the Kobayashi–Hitchin correspondence holds for strictly slope semistable vector bundles which are not polystable.[17] On such vector bundles one may prove the existence of a so-called approximate Hermite–Einstein metric, which is a family ${\displaystyle h_{\varepsilon }}$ of Hermitian metrics for small ${\displaystyle \varepsilon >0}$ such that ${\displaystyle \|\Lambda _{\omega }F(h_{\varepsilon })-\lambda (E)\operatorname {Id} _{E}\|_{L^{2}}<\varepsilon }$ for every ${\displaystyle \varepsilon }$.
• The Kobayashi–Hitchin correspondence has been generalised by Bando–Siu to singular holomorphic vector bundles, otherwise known as reflexive sheaves.[21] This involves defining a notion of singular Hermite–Einstein metrics on such sheaves and has been influential in the developments of singular Kähler–Einstein metrics over singular Fano varieties.
• The correspondence was generalised to the case of Higgs bundles by Hitchin, Carlos Simpson, Donaldson, and Kevin Corlette. Namely Hitchin proved an partial analogue of the Kobayashi–Hitchin correspondence for Higgs bundles over a compact Riemann surface, and Donaldson provided some work on harmonic representations which completed this correspondence. This was then vastly generalised by Simpson to the case of Higgs bundles over arbitrary compact Kähler manifolds, and Corlette proved the corresponding results about harmonic representations in this case. This has come to be known as the nonabelian Hodge correspondence, and has deep relations to mirror symmetry and the P=W conjecture of Tamas Hausel, as well as to integrable systems. The nonabelian Hodge correspondence implies the Kobayashi–Hitchin correspondence for compact Kähler manifolds.
• The correspondence was generalised to Hermite–Einstein metrics on holomorphic principal bundles with reductive structure group, admitting a compatible reduction of structure group to a maximal compact subgroup. Annamalai Ramanathan first defined the notion of a stable principal bundle,[22] and in general the correspondence was proven by Anchouche and Biswas.[23] A version of the correspondence for Higgs-principal bundles is also known.
• David Gieseker introduced a notion of stability, Gieseker stability, which shares many formal properties with slope stability. Gieseker stability asks for inequalities of entire (normalised) Hilbert polynomials for large argument, whereas slope stability asks just for an inequality of the leading order coefficients. Thus Gieseker stability can be seen as a generalisation of slope stability, and indeed there is a chain of implications

slope stable ⇒ Gieseker stable ⇒ Gieseker semistable ⇒ slope semistable.

Gieseker stability is a notion of stability for vector bundles that arises directly out of geometric invariant theory, and has subsequently had significant impact in algebraic geometry, where it is used to form moduli spaces of sheaves.[24] A generalisation of the Kobayashi–Hitchin correspondence was proven for Gieseker stable vector bundles by Conan Leung, who associated to each Gieseker stable vector bundle a so-called almost Hermite–Einstein metric.[25] These are special Hermitian metrics which satisfy a polynomial version of the differential equation defining a Hermite–Einstein metric, and are in fact special classes of approximate Hermite–Einstein metrics.

• In 2001 Álvarez-Cónsul and García-Prada proved a vast generalisation of the Kobayashi–Hitchin correspondence to twisted quiver bundles over compact Kähler manifolds, which are families of holomorphic vector bundles equipped with auxiliary fields and bundle homomorphisms between them. This includes as special cases the regular Kobayashi–Hitchin correspondence, as well as the nonabelian Hodge correspondence and various version of the Kobayashi–Hitchin correspondence for dimensional reductions of the Yang–Mills equations.[26]

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]

• There is a correspondence in Seiberg–Witten theory inspired by the Kobayashi–Hitchin correspondence, which identifies solutions of the Seiberg–Witten equations over a Kähler surface, monopoles, with certain divisors.[29][30] This has been used to compute examples of Seiberg–Witten invariants of four-manifolds and recover results known from Donaldson theory.
• Yau conjectured in 1993 that there should exist a notion of stability for algebraic varieties which would uniquely characterise the existence of Kähler–Einstein metrics on smooth Fano varieties, and that this notion of stability should be an analogue of slope stability of vector bundles.[31] Tian Gang gave a precise definition of such a stability notion, called K-stability, which was rephrased in a purely algebro-geometric way by Donaldson.[32][33] The conjecture that such K-polystable Fano manifolds are in correspondence with Kähler–Einstein metrics was proven by Chen–Donaldson–Sun.[34][35][36]
• Building on the conjecture of Yau, Donaldson conjectured that more generally any smooth K-polystable projective variety should admit a constant scalar curvature Kähler metric. This generalisation of the conjecture for Fano manifolds is known as the Yau–Tian–Donaldson conjecture, and is still open in general. It has been resolved in the case of toric varieties of complex dimension two. Many of the techniques developed to understand the Kobayashi–Hitchin correspondence have been applied to the setting of varieties in order to try and understand the YTD conjecture. Namely the use of the Kähler–Ricci flow as an analogy of the Yang–Mills flow, and of the Calabi functional and K-energy functional in comparison to the Yang–Mills functional and Donaldson functional. The study of optimal degenerations of projective varieties with respect to K-stability has also been heavily inspired by the study of the Harder–Narasimhan filtration of a holomorphic vector bundle, and the singular behaviour of metrics on varieties is studied through analogy with how Hermitian metrics degenerate along the Yang–Mills flow on strictly semistable holomorphic vector bundles.
• The Thomas–Yau conjecture in symplectic geometry proposes a stability condition which should precisely characterise when an isotopy class of Lagrangian submanifolds of a Calabi–Yau manifold admits a special Lagrangian submanifold as a representative.[8] This conjecture can be seen as a direct analogy to the Kobayashi–Hitchin correspondence, where the isotopy class is replaced by a gauge orbit inside the space of Hermitian vector bundles, and the special Lagrangian condition is replaced with the Hermite–Einstein condition. One characterisation of the required stability condition was proposed by Dominic Joyce to come from Bridgeland stability conditions, and a mirror version of the result for the so-called deformed Hermitian Yang–Mills equation has been proven by Gao Chen.[37]