##
**Coherent analytic sheaves.**
*(English)*
Zbl 0537.32001

Grundlehren der Mathematischen Wissenschaften, 265. Berlin etc.: Springer-Verlag. XVIII, 249 p. DM 118.00; $ 46.30 (1984).

This is the long awaited third part of the ”trilogy” on complex analytic geometry by the authors, the previous two parts being ”Analytische Stellenalgebren” (1971; Zbl 0231.32001) and ”Theorie der Steinschen Räume” (1977; Zbl 0379.32001) (referred to as [AS] and [TSS]). The present book contains the general theory of complex analytic spaces. Though it is the third in a series, it is in a large degree self- contained; it includes some material from [TSS], and some results, proved algebraically in [AS], are given geometric proofs here. Only in the last chapter (devoted to the direct image theorem) facts from [TSS] are used.

After the introduction, containing the history of coherent sheaves and a brief description of the contents of the book, the authors collect basic facts about complex spaces in chapter 1. It contains many instructive examples, explanations and warnings. (The authors advise the reader to start reading with chapter 2, and to look into chapter 1 only when necessary, but the reviewer finds it very useful to devote more attention to chapter 1).

Chapter 2 is partially taken from chapter 1 of [TSS], and it refers to [AS]. The Weierstraß division and preparation theorems are proved, finite holomorphic maps are examined (with the exactness of the functor \(f_*\) for a finite map f) and then the Weierstraß isomorphism theorem and the theorem of Oka of the coherence of the sheaf of holomorphic functions on a complex space are proved.

Chapter 3 contains a further study of finite maps (with the result that the coherence is preserved by them), the Rückert Nullstellensatz for coherent sheaves (corresponding to the Hilbert Nullstellensatz in algebraic geometry), relations between torsion sheaves and the openness of finite maps and also the local description of analytic sets in \({\mathbb{C}}^ n\) as branched coverings.

Chapter 4 begins with a detailed study of analytic sets in a complex space and their ideal sheaves. This results in the Rückert Nullstellensatz for ideal sheaves which, for a coherent sheaf of ideals \({\mathcal I}\) in the sheaf of holomorphic functions on a complex space X, says that \(i(N({\mathcal I}))=rad {\mathcal I},\) where \(N({\mathcal I})\) denotes the zero set of \({\mathcal I}\) and \(i(V)\) stands for the (full) sheaf of ideals of an analytic set V in X. Next the authors show that the image of an analytic set under a finite holomorphic map is again analytic and prove a theorem stating the coherence of the sheaf of ideals for any analytic set. As a corollary they obtain a canonical identification of analytic sets and reduced complex subspaces. In the end of the chapter the notion of corank of a coherent sheaf is introduced and is used for showing that a coherent sheaf is locally free outside an analytic set.

Chapter 5 is devoted to the dimension theory of complex spaces. First analytic and algebraic definitions of dimension are compared. The analytic definition says that a complex space X is of dimension n at a point x if n is the minimal number of functions holomorphic in a neighbourhood of x such that \(\{\) \(x\}\) is the set of their common zeros. Algebraically the dimension of X at x is defined as the Chevalley dimension of \({\mathcal O}_ x\). Next active functions are discussed. Among applications there are e.g. open mapping theorems (for maps with discrete fibres and holomorphic functions), local and absolute maximum principles and also the Noether lemma for coherent sheaves saying that in a coherent sheaf an increasingly filtered family of coherent subsheaves has a maximal element on any compact subset.

The main subjects of chapter 6 are the analyticity of the singular locus of a complex space and the normalization of the sheaf of holomorphic functions. The basic notions are the embedding dimension and the sheaf of meromorphic functions. Finally it is shown that the set of non-normal points is analytic and that normal complex spaces are complex spaces with the singular locus of codimension at least two.

In chapter 7 first the Riemann extension theorems for complex manifolds are proved. Then analytic coverings are studied and theorems of integral dependence and of primitive element are proved. As applications various generalizations of the Riemann theorems for locally pure dimensional complex spaces and the Weierstraß convergence theorem for such spaces are obtained. In the last section certain special vector bundles and complex spaces obtained by means of them (e.g. Segre cones) are discussed.

In chapter 8 the coherence of the normalization sheaf of a reduced complex space X and then the existence and uniqueness of the normalization of X are proved. Later topological and analytical properties of the normalization are examined.

Chapter 9 is centred around questions of the irreducibility and connectivity of reduced complex spaces. At the beginning there is a theorem describing the irreducibility of a reduced complex space in eight equivalent ways. Next the global decomposition of a complex space X into irreducible components and its connection with the normalization of X are discussed. Also some place is devoted to proper maps, holomorphically spreadable spaces and questions concerning the local and arcwise connectedness of complex spaces. Later the Remmert-Stein extension theorem for analytic sets is proved. Consequences are the Levy extension theorem for meromorphic functions and theorems of Chow and Hurwitz- Weierstraß on analytic subsets of and meromorphic functions on a complex projective space.

The last chapter 10 is devoted to the proof and applications of the direct image theorem of Grauert saying that the higher direct images of a coherent sheaf under a proper map are coherent. The method used in the proof is of Forster and Knorr. Among many applications there are e.g. examination of regular families of compact complex manifolds with the theorem of the upper-semicontinuity (in the Zariski topology) for dimensions of cohomology groups of fibres of this family with coefficients in restrictions of a vector bundle on the family to the fibres (flat maps are not discussed), the proper mapping theorem saying that the image of an analytic set under a proper map is again analytic, the Stein factorization theorem for proper maps, examination of proper modifications, a detailed study of meromorphic functions on a complex space (with the Thimm-Siegel theorem of algebraic dependence) and the holomorphic reduction of a holomorphically convex space, also known as its petrification. - The book finishes with an annex in which basic facts about sheaves and their coherence are collected.

Generally this book is written with great mastership. It comprises a very large scope of material, proofs are transparent, in the whole text the authors give numerous explanations and examples. In many places there are interesting historical notes. - This book is indispensable for everyone interested in the theory of complex spaces.

After the introduction, containing the history of coherent sheaves and a brief description of the contents of the book, the authors collect basic facts about complex spaces in chapter 1. It contains many instructive examples, explanations and warnings. (The authors advise the reader to start reading with chapter 2, and to look into chapter 1 only when necessary, but the reviewer finds it very useful to devote more attention to chapter 1).

Chapter 2 is partially taken from chapter 1 of [TSS], and it refers to [AS]. The Weierstraß division and preparation theorems are proved, finite holomorphic maps are examined (with the exactness of the functor \(f_*\) for a finite map f) and then the Weierstraß isomorphism theorem and the theorem of Oka of the coherence of the sheaf of holomorphic functions on a complex space are proved.

Chapter 3 contains a further study of finite maps (with the result that the coherence is preserved by them), the Rückert Nullstellensatz for coherent sheaves (corresponding to the Hilbert Nullstellensatz in algebraic geometry), relations between torsion sheaves and the openness of finite maps and also the local description of analytic sets in \({\mathbb{C}}^ n\) as branched coverings.

Chapter 4 begins with a detailed study of analytic sets in a complex space and their ideal sheaves. This results in the Rückert Nullstellensatz for ideal sheaves which, for a coherent sheaf of ideals \({\mathcal I}\) in the sheaf of holomorphic functions on a complex space X, says that \(i(N({\mathcal I}))=rad {\mathcal I},\) where \(N({\mathcal I})\) denotes the zero set of \({\mathcal I}\) and \(i(V)\) stands for the (full) sheaf of ideals of an analytic set V in X. Next the authors show that the image of an analytic set under a finite holomorphic map is again analytic and prove a theorem stating the coherence of the sheaf of ideals for any analytic set. As a corollary they obtain a canonical identification of analytic sets and reduced complex subspaces. In the end of the chapter the notion of corank of a coherent sheaf is introduced and is used for showing that a coherent sheaf is locally free outside an analytic set.

Chapter 5 is devoted to the dimension theory of complex spaces. First analytic and algebraic definitions of dimension are compared. The analytic definition says that a complex space X is of dimension n at a point x if n is the minimal number of functions holomorphic in a neighbourhood of x such that \(\{\) \(x\}\) is the set of their common zeros. Algebraically the dimension of X at x is defined as the Chevalley dimension of \({\mathcal O}_ x\). Next active functions are discussed. Among applications there are e.g. open mapping theorems (for maps with discrete fibres and holomorphic functions), local and absolute maximum principles and also the Noether lemma for coherent sheaves saying that in a coherent sheaf an increasingly filtered family of coherent subsheaves has a maximal element on any compact subset.

The main subjects of chapter 6 are the analyticity of the singular locus of a complex space and the normalization of the sheaf of holomorphic functions. The basic notions are the embedding dimension and the sheaf of meromorphic functions. Finally it is shown that the set of non-normal points is analytic and that normal complex spaces are complex spaces with the singular locus of codimension at least two.

In chapter 7 first the Riemann extension theorems for complex manifolds are proved. Then analytic coverings are studied and theorems of integral dependence and of primitive element are proved. As applications various generalizations of the Riemann theorems for locally pure dimensional complex spaces and the Weierstraß convergence theorem for such spaces are obtained. In the last section certain special vector bundles and complex spaces obtained by means of them (e.g. Segre cones) are discussed.

In chapter 8 the coherence of the normalization sheaf of a reduced complex space X and then the existence and uniqueness of the normalization of X are proved. Later topological and analytical properties of the normalization are examined.

Chapter 9 is centred around questions of the irreducibility and connectivity of reduced complex spaces. At the beginning there is a theorem describing the irreducibility of a reduced complex space in eight equivalent ways. Next the global decomposition of a complex space X into irreducible components and its connection with the normalization of X are discussed. Also some place is devoted to proper maps, holomorphically spreadable spaces and questions concerning the local and arcwise connectedness of complex spaces. Later the Remmert-Stein extension theorem for analytic sets is proved. Consequences are the Levy extension theorem for meromorphic functions and theorems of Chow and Hurwitz- Weierstraß on analytic subsets of and meromorphic functions on a complex projective space.

The last chapter 10 is devoted to the proof and applications of the direct image theorem of Grauert saying that the higher direct images of a coherent sheaf under a proper map are coherent. The method used in the proof is of Forster and Knorr. Among many applications there are e.g. examination of regular families of compact complex manifolds with the theorem of the upper-semicontinuity (in the Zariski topology) for dimensions of cohomology groups of fibres of this family with coefficients in restrictions of a vector bundle on the family to the fibres (flat maps are not discussed), the proper mapping theorem saying that the image of an analytic set under a proper map is again analytic, the Stein factorization theorem for proper maps, examination of proper modifications, a detailed study of meromorphic functions on a complex space (with the Thimm-Siegel theorem of algebraic dependence) and the holomorphic reduction of a holomorphically convex space, also known as its petrification. - The book finishes with an annex in which basic facts about sheaves and their coherence are collected.

Generally this book is written with great mastership. It comprises a very large scope of material, proofs are transparent, in the whole text the authors give numerous explanations and examples. In many places there are interesting historical notes. - This book is indispensable for everyone interested in the theory of complex spaces.

Reviewer: K.Dabrowski

### MSC:

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

32Q99 | Complex manifolds |

32Lxx | Holomorphic fiber spaces |

32B05 | Analytic algebras and generalizations, preparation theorems |

32L05 | Holomorphic bundles and generalizations |

32C15 | Complex spaces |