Lokale Modulräume in der analytischen Geometrie. Band 1 und 2. (Local moduli spaces in analytic geometry. Part 1 and 2).

*(German)*Zbl 0644.32001
Aspects of Mathematics, D2, D3. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn. XIII, XIX, 709 S.; Bd. 1: DM 72.00; Bd. 2: DM 72.00 (1987).

The book is dedicated to the proof of the main results concerning the existence of semiuniversal deformations in analytic geometry. An elaborated, yet unitary framework allows the derivation of all existence statements by means of a common procedure, despite the diversity of the analytical objects involved: isolated singularities, compact complex spaces, holomorphic mappings, coherent sheaves, principal bundles, 1- convex spaces, etc. Essentially, the method goes back to Palamodov’s construction of a (co)tangent complex, (using simplicial resolvents), in his solution of the local moduli problem for compact complex spaces [see for instance: V. P. Palamodov, Russ. Math. Surv. 31, No.3, 129-197 (1976); translation from Usp. Mat. Nauk 31, No.3(189), 129-194 (1976; Zbl 0332.32013)].

Volume one contains holomological and analytical preparations. The categorial setting emphasizes homotopy aspects, due to the merely additive character of the categories of topological vector spaces involved. PO-spaces (which are \({\mathbb{C}}\)-vector spaces with a family \(\| \|_{\lambda}\), \(\lambda\in (0,1)\) of seminorms such that \(\| \|_{\lambda '}\leq \| \|_{\lambda}\) for \(\lambda\) ’\(\leq \lambda)\) are considered in chapter two. An appropriate notion of privileged polycylinders leads to a splitting criteria for certain complexes of sections, to be used later in the case of the tangent complex. The volume ends with a quotient theorem which is the pattern for various existence theorems obtained in volume two. It relies on directness assumption whose verification is quite difficult.

Volume two introduces the (co)tangent complex proving its homotopy invariance and a splitting theorem which guarantees the directness alluded to above.

The results on deformations are given in a very general form, using the notion of graded complex spaces which allows the simultaneous treatment of spaces and modules.

In the case of holomorphic mappings the following theorem is obtained: Let \(f: X\to Y\) be a proper mapping of complex spaces for which the \({\mathcal O}_ Y\)-module \({\mathcal T}^ 1(f,{\mathcal O}_ Y)\) has discrete support. Let \(F\subset Y\) be finite. Then the germ (f,F): (X,f\({}^{- 1}(F))\to (Y,F)\) of f over F has a semiuniversal deformation.

When Y is a simple point, this gives the existence theorem for compact complex spaces.

The technical apparatus which in the end leads to such geometrically palpable results is quite demanding and unity of procedure requires higher levels of abstraction.

A few external references remain necessary for logical completeness, while for motivations, intuitive background and early history of the subject, a book such as K. Kodaira’s “Complex manifolds and deformations of complex structures”, Springer Verlag (1986; Zbl 0581.32012) might prove helpfull.

Volume one contains holomological and analytical preparations. The categorial setting emphasizes homotopy aspects, due to the merely additive character of the categories of topological vector spaces involved. PO-spaces (which are \({\mathbb{C}}\)-vector spaces with a family \(\| \|_{\lambda}\), \(\lambda\in (0,1)\) of seminorms such that \(\| \|_{\lambda '}\leq \| \|_{\lambda}\) for \(\lambda\) ’\(\leq \lambda)\) are considered in chapter two. An appropriate notion of privileged polycylinders leads to a splitting criteria for certain complexes of sections, to be used later in the case of the tangent complex. The volume ends with a quotient theorem which is the pattern for various existence theorems obtained in volume two. It relies on directness assumption whose verification is quite difficult.

Volume two introduces the (co)tangent complex proving its homotopy invariance and a splitting theorem which guarantees the directness alluded to above.

The results on deformations are given in a very general form, using the notion of graded complex spaces which allows the simultaneous treatment of spaces and modules.

In the case of holomorphic mappings the following theorem is obtained: Let \(f: X\to Y\) be a proper mapping of complex spaces for which the \({\mathcal O}_ Y\)-module \({\mathcal T}^ 1(f,{\mathcal O}_ Y)\) has discrete support. Let \(F\subset Y\) be finite. Then the germ (f,F): (X,f\({}^{- 1}(F))\to (Y,F)\) of f over F has a semiuniversal deformation.

When Y is a simple point, this gives the existence theorem for compact complex spaces.

The technical apparatus which in the end leads to such geometrically palpable results is quite demanding and unity of procedure requires higher levels of abstraction.

A few external references remain necessary for logical completeness, while for motivations, intuitive background and early history of the subject, a book such as K. Kodaira’s “Complex manifolds and deformations of complex structures”, Springer Verlag (1986; Zbl 0581.32012) might prove helpfull.

Reviewer: N.Mihalache

##### MSC:

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

32G10 | Deformations of submanifolds and subspaces |

32G13 | Complex-analytic moduli problems |

32S30 | Deformations of complex singularities; vanishing cycles |

32G05 | Deformations of complex structures |