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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.
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)