The Kobayashi-Hitchin correspondence.

*(English)*Zbl 0849.32020
Singapore: World Scientific. viii, 254 p. (1995).

This book gives a detailed and complete treatment of the Kobayashi-Hitchin correspondence. Roughly speaking this states a 1-1-correspondence between stable holomorphic structures on complex vector bundles over a compact complex manifold and Hermite-Einstein metrics on such bundles.

Stability was a concept coming from geometric invariant theory. Here, the so-called Mumford-Takemoto stability is considered, which originally is related to a projective embedding of the variety. It is quite natural to define this concept with respect to a given Kähler metric (a projective embedding is a special case). In this book the more general case of stability with respect to a Gauduchon metric is treated. A Gauduchon metric is a Hermitian metric whose associated (1,1)-form \(\omega\) satisfies \(\partial \overline \partial (\omega^{n-1}) = 0\) \((n\) the dimension of the variety) (the Kähler condition would be the stronger condition \(d \omega = 0)\).

This is a quite general concept, since on a compact complex manifold \(X\) each class of conformal equivalent Hermitian metrics contains a Gauduchon metric (unique up to rescaling, if \(X\) is connected and \(n \geq 2)\) [P. Gauduchon, Math. Ann. 267, 495-518 (1984; Zbl 0536.53066)]. The book contains a proof of this result. Integrating the Chern-form of a Hermitian holomorphic line bundle against \(\omega^{g-1}\) gives a degree map \(\text{Pic} (X) \to \mathbb{R}\) (in general it is not a topological invariant as it is known in the Kähler case!) Having a degree map it allows to define the slope of torsion free coherent analytic sheaves \({\mathcal F}\) on \(X\) as \(\mu ({\mathcal F}) = \deg (\text{det} ({\mathcal F}))/r\), where \(r = rk ({\mathcal F})\) and where \(\text{det} ({\mathcal F})\) is the bidual of \(\wedge^r {\mathcal F}\). Then \({\mathcal F}\) is called (semi-)stable if for every coherent analytic subsheaf \({\mathcal G} \subset {\mathcal F}\) with \(0 < rk ({\mathcal G} < rk ({\mathcal F})\) one has \(\mu ({\mathcal G}) < (\leq) \mu_g ({\mathcal F})\).

Now let \(E\) be a smooth complex vector bundle on the compact complex manifold \(X,g\) a Gauduchon metric on \(X\) and \(h_0\) a Hermitian metric on \(E\). A stable holomorphic structure on \(E\) is a holomorphic structure \(\overline \delta\) (i.e. an integrable (0,1)-semiconnection) such that the associated coherent analytic sheaf \((= \text{Ker} (\overline \delta))\) is stable.

A Hermite-Einstein connection on \((Eh_0)\) is a unitary connection on \((E,h_0)\) whose curvature tensor \(F\) is of type (1,1) and satisfies \(i\wedge F=\text{const.} Id\) (where \(\wedge\) is the contraction with the (1,1)-form \(\omega\) associated to the metric \(g)\).

It was known since some time that the (0,1)-part of a Hermite-Einstein connection is a polystable holomorphic structure on \(E\) (i.e. a direct sum of stable holomorphic structures) and therefore that irreducible Hermite-Einstein connections induce stable holomorphic structures on \(E\). It was a long standing conjecture that the converse statement is also true. The introduction of the book gives a short account on the history of this circle of problems.

The first two chapters then introduce the basic definitions and results on holomorphic structures, Gauduchon metrics, connections, stability, Hermite-Einstein connections.

The heart of the book is Chapter 3 and Chapter 4. Chapter 3 deals with the problem of existence of Hermite-Einstein metrics in stable bundles, it gives a complete, self-containing proof (taking into account the appendices, where the necessary technical tools from analysis are gathered) of the theorem of existence in full generality. Thus it states a bijection between irreducible Hermite-Einstein connections modulo unitary gauge equivalence and stable holomorhic structure modulo gauge equivalence.

In Chapter 4 this correspondence is considered from the point of view of moduli spaces. It is shown that the set of unitary gauge equivalence classes of irreducible Hermite-Einstein connections has the structure of a course moduli space \({\mathcal M}_g^{HE} (E,h_0)\) which has the quality of a finite dimensional real analytical space (which might be non-reduced) and that the set of gauge equivalence classes of stable holomorphic structures has the structure of a course moduli space \({\mathcal M}_g^{st} (E)\) which has the quality of a finite dimensional complex analytic space (non-reduced in general) and finally it is shown that the Kobayashi-Hitchin correspondence is an isomorphism of the underlying real analytic spaces. As last section in chapter 4 it is shown that – using the Kobayashi-Hitchin correspondence – moduli spaces of instantons on compact complex surfaces can be identified with moduli spaces of stable holomorphic bundles. This fact was extensively used in the differential topology of 4-manifolds, studied via Donaldson invariants.

In Chapter 5 the openness of the stability property is shown, roughly stating that small perturbations of the complex structure on \(X\), the metric on \(X\) and the holomorphic structure on \(E\) preserve stability. Then for compact complex surfaces the dependence on the base metric is discussed.

Furthermore, the \(L^2\)-metric on the (smooth part of the) moduli space is studied, it is shown that the associated (1,1)-form \(\Omega\) is \(\partial \overline \partial\)-closed. Finally, a proof of Bogomolov’s theorem on surfaces of type \(VII_0\) is given. Chapter 6 treats examples of moduli spaces (on complex surfaces).

The last Chapter 7 collects as appendices necessary technical tools from analysis, Hermitian geometry and Banach manifolds.

This very well written book is the first systematic and self-contained book on the subject. The non-Kähler context is treated in this generality for the first time and it becomes evident that it is quite natural to state and proof the Kobayashi-Hitchin correspondence in this context. Also the careful study of this correspondence as isomorphism of moduli spaces (instead of merely a bijection) is remarkable. The book contains an extensive bibliography. I think it will become a standard reference for the Kobayashi-Hitchin correspondence and related topics.

Stability was a concept coming from geometric invariant theory. Here, the so-called Mumford-Takemoto stability is considered, which originally is related to a projective embedding of the variety. It is quite natural to define this concept with respect to a given Kähler metric (a projective embedding is a special case). In this book the more general case of stability with respect to a Gauduchon metric is treated. A Gauduchon metric is a Hermitian metric whose associated (1,1)-form \(\omega\) satisfies \(\partial \overline \partial (\omega^{n-1}) = 0\) \((n\) the dimension of the variety) (the Kähler condition would be the stronger condition \(d \omega = 0)\).

This is a quite general concept, since on a compact complex manifold \(X\) each class of conformal equivalent Hermitian metrics contains a Gauduchon metric (unique up to rescaling, if \(X\) is connected and \(n \geq 2)\) [P. Gauduchon, Math. Ann. 267, 495-518 (1984; Zbl 0536.53066)]. The book contains a proof of this result. Integrating the Chern-form of a Hermitian holomorphic line bundle against \(\omega^{g-1}\) gives a degree map \(\text{Pic} (X) \to \mathbb{R}\) (in general it is not a topological invariant as it is known in the Kähler case!) Having a degree map it allows to define the slope of torsion free coherent analytic sheaves \({\mathcal F}\) on \(X\) as \(\mu ({\mathcal F}) = \deg (\text{det} ({\mathcal F}))/r\), where \(r = rk ({\mathcal F})\) and where \(\text{det} ({\mathcal F})\) is the bidual of \(\wedge^r {\mathcal F}\). Then \({\mathcal F}\) is called (semi-)stable if for every coherent analytic subsheaf \({\mathcal G} \subset {\mathcal F}\) with \(0 < rk ({\mathcal G} < rk ({\mathcal F})\) one has \(\mu ({\mathcal G}) < (\leq) \mu_g ({\mathcal F})\).

Now let \(E\) be a smooth complex vector bundle on the compact complex manifold \(X,g\) a Gauduchon metric on \(X\) and \(h_0\) a Hermitian metric on \(E\). A stable holomorphic structure on \(E\) is a holomorphic structure \(\overline \delta\) (i.e. an integrable (0,1)-semiconnection) such that the associated coherent analytic sheaf \((= \text{Ker} (\overline \delta))\) is stable.

A Hermite-Einstein connection on \((Eh_0)\) is a unitary connection on \((E,h_0)\) whose curvature tensor \(F\) is of type (1,1) and satisfies \(i\wedge F=\text{const.} Id\) (where \(\wedge\) is the contraction with the (1,1)-form \(\omega\) associated to the metric \(g)\).

It was known since some time that the (0,1)-part of a Hermite-Einstein connection is a polystable holomorphic structure on \(E\) (i.e. a direct sum of stable holomorphic structures) and therefore that irreducible Hermite-Einstein connections induce stable holomorphic structures on \(E\). It was a long standing conjecture that the converse statement is also true. The introduction of the book gives a short account on the history of this circle of problems.

The first two chapters then introduce the basic definitions and results on holomorphic structures, Gauduchon metrics, connections, stability, Hermite-Einstein connections.

The heart of the book is Chapter 3 and Chapter 4. Chapter 3 deals with the problem of existence of Hermite-Einstein metrics in stable bundles, it gives a complete, self-containing proof (taking into account the appendices, where the necessary technical tools from analysis are gathered) of the theorem of existence in full generality. Thus it states a bijection between irreducible Hermite-Einstein connections modulo unitary gauge equivalence and stable holomorhic structure modulo gauge equivalence.

In Chapter 4 this correspondence is considered from the point of view of moduli spaces. It is shown that the set of unitary gauge equivalence classes of irreducible Hermite-Einstein connections has the structure of a course moduli space \({\mathcal M}_g^{HE} (E,h_0)\) which has the quality of a finite dimensional real analytical space (which might be non-reduced) and that the set of gauge equivalence classes of stable holomorphic structures has the structure of a course moduli space \({\mathcal M}_g^{st} (E)\) which has the quality of a finite dimensional complex analytic space (non-reduced in general) and finally it is shown that the Kobayashi-Hitchin correspondence is an isomorphism of the underlying real analytic spaces. As last section in chapter 4 it is shown that – using the Kobayashi-Hitchin correspondence – moduli spaces of instantons on compact complex surfaces can be identified with moduli spaces of stable holomorphic bundles. This fact was extensively used in the differential topology of 4-manifolds, studied via Donaldson invariants.

In Chapter 5 the openness of the stability property is shown, roughly stating that small perturbations of the complex structure on \(X\), the metric on \(X\) and the holomorphic structure on \(E\) preserve stability. Then for compact complex surfaces the dependence on the base metric is discussed.

Furthermore, the \(L^2\)-metric on the (smooth part of the) moduli space is studied, it is shown that the associated (1,1)-form \(\Omega\) is \(\partial \overline \partial\)-closed. Finally, a proof of Bogomolov’s theorem on surfaces of type \(VII_0\) is given. Chapter 6 treats examples of moduli spaces (on complex surfaces).

The last Chapter 7 collects as appendices necessary technical tools from analysis, Hermitian geometry and Banach manifolds.

This very well written book is the first systematic and self-contained book on the subject. The non-Kähler context is treated in this generality for the first time and it becomes evident that it is quite natural to state and proof the Kobayashi-Hitchin correspondence in this context. Also the careful study of this correspondence as isomorphism of moduli spaces (instead of merely a bijection) is remarkable. The book contains an extensive bibliography. I think it will become a standard reference for the Kobayashi-Hitchin correspondence and related topics.

Reviewer: H.Kurke (Berlin)

##### MSC:

32Q20 | Kähler-Einstein manifolds |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |

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