Deformations of coherent analytic sheaves with compact supports.

*(English)*Zbl 0526.32021
Mem. Am. Math. Soc. 238, 155 p. (1981).

This paper is devoted to the construction of a semi-universal deformation of any coherent analytic sheaf on a complex space which has compact support. The procedure is constructive and elementary. It uses the power series method and the division and extension theory of ideals in power series rings developed by H.Grauert [Invent. Math. 15, 171-198 (1972; Zbl 0237.32011)].

This procedure has a number of important special features involving new techniques. Some of these features are: (i) the construction of a partial sheaf system resolution where maps for set-inclusions have directions opposite to those in a sheaf system resolution used for the proof of the direct image theorem, (ii) the use of partially privileged boundaries and Hörmander’s \(L^ 2\) estimates of \({\bar \partial}\) to avoid the need of secondary smoothing, (iii) the establishment of bounds independent of \(e\) for extending resolutions of kernels of sheaf-homomorphisms from the \(e^{th}\) to the \((e+1)^{st}\) step and for lifting sections through sheaf-homomorphisms and expressing cocycles as coboundaries at the \(e^{th}\) step.

This procedure has a number of important special features involving new techniques. Some of these features are: (i) the construction of a partial sheaf system resolution where maps for set-inclusions have directions opposite to those in a sheaf system resolution used for the proof of the direct image theorem, (ii) the use of partially privileged boundaries and Hörmander’s \(L^ 2\) estimates of \({\bar \partial}\) to avoid the need of secondary smoothing, (iii) the establishment of bounds independent of \(e\) for extending resolutions of kernels of sheaf-homomorphisms from the \(e^{th}\) to the \((e+1)^{st}\) step and for lifting sections through sheaf-homomorphisms and expressing cocycles as coboundaries at the \(e^{th}\) step.

##### MSC:

32L10 | Sheaves and cohomology of sections of holomorphic vector bundles, general results |

32G05 | Deformations of complex structures |

32C35 | Analytic sheaves and cohomology groups |

13F25 | Formal power series rings |