##
**The universal Kobayashi-Hitchin correspondence on Hermitian manifolds.**
*(English)*
Zbl 1103.53014

Mem. Am. Math. Soc. 863, 97 p. (2006).

In a nutshell, the classical and famous Kobayashi-Hitchin correspondence relates a global extrinsic algebraic condition to the existence of an intrinsic object which is the solution of a nonlinear PDE. More precisely, it states that over a compact complex manifold \(X\), the polystability of a holomorphic vector bundle \(E\) is equivalent to the existence of a Hermitian-Einstein metric, i.e., a smooth metric \(h\) for \(E\) solving the equation

\[ \sqrt{-1} \Lambda_g F_h = \mu_g(E)\text{Id}_E \in C^{\infty}(M, \text{End}(E)) \]

where \(F_h\) stands for the curvature of \(h\) and \(\mu_g(E)\) is a constant, called the slope of \(E\). The notion of stability depends on the choice of a Gauduchon metric \(g\) (for which one can naturally associate the contraction operator \(\Lambda_g\) on \((1,1)\)-forms) when one just assumes that \(X\) is complex compact and plays the role of a polarization for algebraic manifolds (in that case, the slope of \(E\) is given by its degree divided by its rank). One says that \(E\) is stable if for any non-trivial subsheaf \(F \subset E\) with torsion free quotient, one has the slope-inequality

\[ \mu_g(F) < \mu_g(E), \]

and that \(E\) is polystable if it is a direct sum of stable bundles of the same slope.

This deep result has been first conjectured by Kobayashi and Hitchin around 1980. The simpler implication (existence of a solution \(\Rightarrow\) stability) was proved by Kobayashi and independently Lübke in the early 80’s in the algebraic setting. The difficult implication (stability \(\Rightarrow\) existence of solution) was proved by Donaldson for algebraic manifolds in a series of fundamental papers. Uhlenbeck and Yau have extended these results to Kähler manifolds, Buchdahl to Hermitian surfaces, and finally Li and Yau to arbitrary Hermitian manifolds.

Why is Kobayashi-Hitchin correspondence so important in complex geometry? If we are given a differentiable vector bundle \({\mathrm E}\) over \(X\), it is a fundamental problem to classify all holomophic structures on \(\mathrm{E}\) which induce a fixed holomorphic structure on \(\det(\mathrm{E})\) (modulo the group \(\Gamma(X,\text{SL}(\mathrm{E}))\) of automorphisms of determinant 1). Let us now fix a metric \(\mathrm{h}\) on \(\mathrm{E}\). The Kobayashi-Hitchin correspondence gives an isomorphism of moduli space between the moduli space of stable holomorphic structures with fixed determinant and the moduli of irreducible integrable Hermitian-Einstein connections \(A\) with fixed determinant on \((\mathrm{E,h})\). In the case of surfaces, the study of this latter moduli (identified as the moduli of ASD connections) has led to major consequences in topology and for physicists via gauge theory.

One can try to extend the Kobayashi-Hitchin correspondence in different directions in order to understand more general moduli problems. The most natural is to consider extra data, for instance Higgs pairs, holomorphic pairs or Witten triples. Another way is to consider very general base manifolds, like compact complex manifolds that come with a Gauduchon metric (the case of open manifolds is still a topic of active research). In this book, the authors investigate the following very general case. They consider an exact sequence

\[ {1} \rightarrow G \rightarrow \widehat{G} \rightarrow G_0 \rightarrow {1} \]

of complex reductive groups and choose a \(\widehat{G}\)-bundle \(\widehat{\mathrm{Q}}\) on a compact complex manifold \(X\) and a holomorphic action \(\widehat{\alpha}:\widehat{G} \times F \rightarrow F\) on a Kählerian manifold \(F\). Let \(\mathrm{Q}_0=\widehat{\mathrm{Q}}/G\) be the associated \(G_0\) bundle and \(\mathrm{E}\) the associated \(F\)-bundle, \(\mathrm{E}=\widehat{\mathrm{Q}}\times_{\widehat{\alpha}}F\). Fix a holomorphic structure \(\mathcal{Q}_0\) on \(\mathrm{Q}_0\). Now, the “universal” classification problems they deal with, stands up as

Classify the oriented pairs \((\widehat{\mathcal{Q}},\phi)\) of type \((\widehat{\mathrm{Q}},\widehat{\alpha},\mathcal{Q}_0)\) where \(\widehat{\mathcal{Q}}\) is a holomorphic structure on \(\widehat{\mathrm{Q}}\) inducing the fixed structure \(\mathcal{Q}_0\) on \(\mathrm{Q}_0\) and \(\phi \in \Gamma(X,E)\) is holomorphic with respect to the holomorphic structure induced by \(\widehat{\mathcal{Q}}\) on \(\mathrm{E}\), modulo the gauge group \(\mathcal{G}=\operatorname{Aut}_{\mathrm{Q}_0}(\widehat{\mathrm{Q}})\).

As fas as the reviewer knows, this moduli problem overlaps with all the previously studied moduli problems in literature (Witten triples, Higgs bundles, quiver structures, non-oriented or oriented pairs,…).

Structure of the paper: The first chapter, “Finite dimensional Kobayashi-Hitchin correspondence”, gives a numerical characterization of the polystable orbits with respect to a holomorphic Hamiltonian action \((\alpha,\mu)\) satisfying a certain condition, called energy completeness condition. Let us emphasize this interesting notion [see for details A. Teleman, Int. J. Math. 15, No. 2, 183–209 (2004; Zbl 1089.53058)]. For a holomorphic action \(\alpha : G \times M \rightarrow M\) of a complex reductive group \(G\) on a complex manifold \(M\), and a moment map \(\mu\) for the induced \(K\)-action (\(K\) is a maximal compact subgroup of \(G\)) on \((M,\omega_g)\) where \(\omega_g\) is a \(K\)-invariant Kähler metric, one can define the energy \[ E_g(\{ e^{ts}x \}_{t\geq t_0})=\lim_{t\rightarrow +\infty}\langle\mu(e^{ts}x),-is\rangle - \langle\mu(e^{t_0s}x),-is\rangle \] for \(x\in M\) and \(s\in \text{Lie}(K)^*\). The Hamiltonian triple \((K,g,\mu)\) is said of energy complete if for all \(x\in M\) and \(s\in \text{Lie}(K)^*\), one has

\[ E_g(\{ e^{ts}x \}_{t})< \infty \Rightarrow \lim_{t\rightarrow \infty} \{ e^{ts}x \} \text{ exists in } M. \]

This is one of the key points of the whole paper, since under this condition, the authors prove an equivalent of the classical Hilbert-Mumford criterion in G.I.T but for non algebraic non compact structures. The method used here is a continuity method that serves as a toy model for the infinite dimensional gauge problem. In particular, this implies that the symplectic semistability of \(x\in M\) (i.e., the existence of a zero for the moment map \(\mu\) in the closure of the \(G\)-orbit of \(x\)) is equivalent to \[ \lim_{t\rightarrow +\infty}\langle\mu(e^{ts}x),-is\rangle \geq 0 \] for all \(s \in \text{Lie}(K)^*\) for all maximal compact subgroups \(K \subset G\). An analog of this result for polystability is proved in details.

In the second chapter, the “universal” moduli problem is stated and notions of stability, semistability and polystability are introduced for universal oriented pairs. The main difficulty here is to define a notion of degree for the considered objects in a non Kählerian framework (in particular it is not anymore a topological invariant).

The two following chapters are dedicated to prove the Kobayashi-Hitchin correspondence. As we have stressed, note that for the difficult implication the proof is based on the lines of the Uhlenbeck-Yau’s continuity method, i.e., by perturbing the Hermitian-Einstein equation.

Finally, one third of the paper is dedicated to some nice applications. The authors derive a correspondence for oriented holomorphic principal bundles and explain how the moduli of oriented connections is related to the extended Witten conjecture (i.e., that Donaldson invariants and Seiberg-Witten invariants are related for non-simply connected 4-manifolds). Using the correspondence, the authors exhibit a canonical Hermitian metric on the smooth part of the moduli spaces of oriented connections and on another hand, of moduli spaces of oriented holomorphic pairs. For both moduli, they prove that the associated \((1,1)\)-form \(\Omega\) is Kähler if the base manifold has a semi-Kähler metric \(\omega_g\) (i.e., \(d(\omega^{n-1}_g)=0\)). Moreover it is shown that \(\partial \bar \partial \Omega=0\) if \(g\) is Gauduchon. Of course, such a condition on \(\Omega\) can be understood as a characterization of the geometry of these moduli spaces.

The authors also investigate the case of Douady Quot spaces, i.e., the Quot spaces of quotients of a fixed holomorphic bundle with locally free kernel of a fixed differentiable type. These spaces are closely related to Douady spaces of effective divisors representing \(PD(m)\) for a fixed \(m \in H^2(X,\mathbb{Z})\). For such spaces, there exists a canonical \((1,1)\)-form with the same properties as previously depending on the metric \(g\) on \(X\). Finally, a section is dedicated to non-Abelian Seiberg-Witten theory for Gauduchon surfaces for which the geometry of the moduli spaces of non-Abelian monopoles is described via the moduli spaces of oriented rank 2 holomorphic pairs. This part is in fact related to the “cobordism strategy” for proving the Witten conjecture for arbitrary simple type manifolds and illustrate why extending the Kobayashi-Hitchin correspondence to an arbitrary Hermitian manifold has some interest for geometers.

Clearly this paper is intended for specialists of the subject and the previous book of the authors [The Kobayashi-Hitchin correspondence. Singapore: World Scientific (1995; Zbl 0849.32020)] is certainly recommended for a first approach on the Kobayashi-Hitchin correspondence. Despite the technicalities of the proofs (due to the non-Kählerian setting) and sometimes some confusing notations, the paper is well written and gives some very powerful results on the moduli spaces of the considered objects.

\[ \sqrt{-1} \Lambda_g F_h = \mu_g(E)\text{Id}_E \in C^{\infty}(M, \text{End}(E)) \]

where \(F_h\) stands for the curvature of \(h\) and \(\mu_g(E)\) is a constant, called the slope of \(E\). The notion of stability depends on the choice of a Gauduchon metric \(g\) (for which one can naturally associate the contraction operator \(\Lambda_g\) on \((1,1)\)-forms) when one just assumes that \(X\) is complex compact and plays the role of a polarization for algebraic manifolds (in that case, the slope of \(E\) is given by its degree divided by its rank). One says that \(E\) is stable if for any non-trivial subsheaf \(F \subset E\) with torsion free quotient, one has the slope-inequality

\[ \mu_g(F) < \mu_g(E), \]

and that \(E\) is polystable if it is a direct sum of stable bundles of the same slope.

This deep result has been first conjectured by Kobayashi and Hitchin around 1980. The simpler implication (existence of a solution \(\Rightarrow\) stability) was proved by Kobayashi and independently Lübke in the early 80’s in the algebraic setting. The difficult implication (stability \(\Rightarrow\) existence of solution) was proved by Donaldson for algebraic manifolds in a series of fundamental papers. Uhlenbeck and Yau have extended these results to Kähler manifolds, Buchdahl to Hermitian surfaces, and finally Li and Yau to arbitrary Hermitian manifolds.

Why is Kobayashi-Hitchin correspondence so important in complex geometry? If we are given a differentiable vector bundle \({\mathrm E}\) over \(X\), it is a fundamental problem to classify all holomophic structures on \(\mathrm{E}\) which induce a fixed holomorphic structure on \(\det(\mathrm{E})\) (modulo the group \(\Gamma(X,\text{SL}(\mathrm{E}))\) of automorphisms of determinant 1). Let us now fix a metric \(\mathrm{h}\) on \(\mathrm{E}\). The Kobayashi-Hitchin correspondence gives an isomorphism of moduli space between the moduli space of stable holomorphic structures with fixed determinant and the moduli of irreducible integrable Hermitian-Einstein connections \(A\) with fixed determinant on \((\mathrm{E,h})\). In the case of surfaces, the study of this latter moduli (identified as the moduli of ASD connections) has led to major consequences in topology and for physicists via gauge theory.

One can try to extend the Kobayashi-Hitchin correspondence in different directions in order to understand more general moduli problems. The most natural is to consider extra data, for instance Higgs pairs, holomorphic pairs or Witten triples. Another way is to consider very general base manifolds, like compact complex manifolds that come with a Gauduchon metric (the case of open manifolds is still a topic of active research). In this book, the authors investigate the following very general case. They consider an exact sequence

\[ {1} \rightarrow G \rightarrow \widehat{G} \rightarrow G_0 \rightarrow {1} \]

of complex reductive groups and choose a \(\widehat{G}\)-bundle \(\widehat{\mathrm{Q}}\) on a compact complex manifold \(X\) and a holomorphic action \(\widehat{\alpha}:\widehat{G} \times F \rightarrow F\) on a Kählerian manifold \(F\). Let \(\mathrm{Q}_0=\widehat{\mathrm{Q}}/G\) be the associated \(G_0\) bundle and \(\mathrm{E}\) the associated \(F\)-bundle, \(\mathrm{E}=\widehat{\mathrm{Q}}\times_{\widehat{\alpha}}F\). Fix a holomorphic structure \(\mathcal{Q}_0\) on \(\mathrm{Q}_0\). Now, the “universal” classification problems they deal with, stands up as

Classify the oriented pairs \((\widehat{\mathcal{Q}},\phi)\) of type \((\widehat{\mathrm{Q}},\widehat{\alpha},\mathcal{Q}_0)\) where \(\widehat{\mathcal{Q}}\) is a holomorphic structure on \(\widehat{\mathrm{Q}}\) inducing the fixed structure \(\mathcal{Q}_0\) on \(\mathrm{Q}_0\) and \(\phi \in \Gamma(X,E)\) is holomorphic with respect to the holomorphic structure induced by \(\widehat{\mathcal{Q}}\) on \(\mathrm{E}\), modulo the gauge group \(\mathcal{G}=\operatorname{Aut}_{\mathrm{Q}_0}(\widehat{\mathrm{Q}})\).

As fas as the reviewer knows, this moduli problem overlaps with all the previously studied moduli problems in literature (Witten triples, Higgs bundles, quiver structures, non-oriented or oriented pairs,…).

Structure of the paper: The first chapter, “Finite dimensional Kobayashi-Hitchin correspondence”, gives a numerical characterization of the polystable orbits with respect to a holomorphic Hamiltonian action \((\alpha,\mu)\) satisfying a certain condition, called energy completeness condition. Let us emphasize this interesting notion [see for details A. Teleman, Int. J. Math. 15, No. 2, 183–209 (2004; Zbl 1089.53058)]. For a holomorphic action \(\alpha : G \times M \rightarrow M\) of a complex reductive group \(G\) on a complex manifold \(M\), and a moment map \(\mu\) for the induced \(K\)-action (\(K\) is a maximal compact subgroup of \(G\)) on \((M,\omega_g)\) where \(\omega_g\) is a \(K\)-invariant Kähler metric, one can define the energy \[ E_g(\{ e^{ts}x \}_{t\geq t_0})=\lim_{t\rightarrow +\infty}\langle\mu(e^{ts}x),-is\rangle - \langle\mu(e^{t_0s}x),-is\rangle \] for \(x\in M\) and \(s\in \text{Lie}(K)^*\). The Hamiltonian triple \((K,g,\mu)\) is said of energy complete if for all \(x\in M\) and \(s\in \text{Lie}(K)^*\), one has

\[ E_g(\{ e^{ts}x \}_{t})< \infty \Rightarrow \lim_{t\rightarrow \infty} \{ e^{ts}x \} \text{ exists in } M. \]

This is one of the key points of the whole paper, since under this condition, the authors prove an equivalent of the classical Hilbert-Mumford criterion in G.I.T but for non algebraic non compact structures. The method used here is a continuity method that serves as a toy model for the infinite dimensional gauge problem. In particular, this implies that the symplectic semistability of \(x\in M\) (i.e., the existence of a zero for the moment map \(\mu\) in the closure of the \(G\)-orbit of \(x\)) is equivalent to \[ \lim_{t\rightarrow +\infty}\langle\mu(e^{ts}x),-is\rangle \geq 0 \] for all \(s \in \text{Lie}(K)^*\) for all maximal compact subgroups \(K \subset G\). An analog of this result for polystability is proved in details.

In the second chapter, the “universal” moduli problem is stated and notions of stability, semistability and polystability are introduced for universal oriented pairs. The main difficulty here is to define a notion of degree for the considered objects in a non Kählerian framework (in particular it is not anymore a topological invariant).

The two following chapters are dedicated to prove the Kobayashi-Hitchin correspondence. As we have stressed, note that for the difficult implication the proof is based on the lines of the Uhlenbeck-Yau’s continuity method, i.e., by perturbing the Hermitian-Einstein equation.

Finally, one third of the paper is dedicated to some nice applications. The authors derive a correspondence for oriented holomorphic principal bundles and explain how the moduli of oriented connections is related to the extended Witten conjecture (i.e., that Donaldson invariants and Seiberg-Witten invariants are related for non-simply connected 4-manifolds). Using the correspondence, the authors exhibit a canonical Hermitian metric on the smooth part of the moduli spaces of oriented connections and on another hand, of moduli spaces of oriented holomorphic pairs. For both moduli, they prove that the associated \((1,1)\)-form \(\Omega\) is Kähler if the base manifold has a semi-Kähler metric \(\omega_g\) (i.e., \(d(\omega^{n-1}_g)=0\)). Moreover it is shown that \(\partial \bar \partial \Omega=0\) if \(g\) is Gauduchon. Of course, such a condition on \(\Omega\) can be understood as a characterization of the geometry of these moduli spaces.

The authors also investigate the case of Douady Quot spaces, i.e., the Quot spaces of quotients of a fixed holomorphic bundle with locally free kernel of a fixed differentiable type. These spaces are closely related to Douady spaces of effective divisors representing \(PD(m)\) for a fixed \(m \in H^2(X,\mathbb{Z})\). For such spaces, there exists a canonical \((1,1)\)-form with the same properties as previously depending on the metric \(g\) on \(X\). Finally, a section is dedicated to non-Abelian Seiberg-Witten theory for Gauduchon surfaces for which the geometry of the moduli spaces of non-Abelian monopoles is described via the moduli spaces of oriented rank 2 holomorphic pairs. This part is in fact related to the “cobordism strategy” for proving the Witten conjecture for arbitrary simple type manifolds and illustrate why extending the Kobayashi-Hitchin correspondence to an arbitrary Hermitian manifold has some interest for geometers.

Clearly this paper is intended for specialists of the subject and the previous book of the authors [The Kobayashi-Hitchin correspondence. Singapore: World Scientific (1995; Zbl 0849.32020)] is certainly recommended for a first approach on the Kobayashi-Hitchin correspondence. Despite the technicalities of the proofs (due to the non-Kählerian setting) and sometimes some confusing notations, the paper is well written and gives some very powerful results on the moduli spaces of the considered objects.

Reviewer: Julien Keller (London)

### MSC:

53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |

58D27 | Moduli problems for differential geometric structures |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

53D20 | Momentum maps; symplectic reduction |

32L05 | Holomorphic bundles and generalizations |

32M05 | Complex Lie groups, group actions on complex spaces |