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Twisted Picard-Lefschetz formulas. (English. Russian original) Zbl 0665.32011
Funct. Anal. Appl. 22, No. 1, 10-18 (1988); translation from Funkts. Anal. Prilozh. 22, No. 1, 12-22 (1988).
Let f: ($${\mathbb{C}}^ n,0)\to ({\mathbb{C}},0)$$ be the germ of a holomorphic function defining an isolated hypersurface singularity of multiplicity $$\mu$$ in $${\mathbb{C}}^ n$$. Furthermore, let F: ($${\mathbb{C}}^ n\times {\mathbb{C}}^{\mu},0\times 0)\to ({\mathbb{C}},0)$$ be a (mini-)versal deformation of f, $$F_{\lambda}:=F(-,\lambda)$$, and $$V_{\lambda}:=F_{\lambda}^{-1}(0)$$ for any $$\lambda\in {\mathbb{C}}$$. The discriminant of the versal deformation $$(V_{\lambda})_{\lambda \in {\mathbb{C}}^{\mu}}$$ of the given hypersurface singularity is defined to be the locus $$\Delta \subset {\mathbb{C}}^{\mu}$$ of the parameters $$\lambda$$ for which the hypersurface $$V_{\lambda}$$ is singular. Then one has a natural representation of the fundamental group $$\pi_ 1({\mathbb{C}}^{\mu}\setminus \Delta)$$ as a linear transformation group of the intermediate homology groups $$H_{n-1}(V_{\lambda},{\mathbb{Z}})\cong {\mathbb{Z}}^{\mu}$$, $$\lambda \in {\mathbb{C}}^{\mu}$$, just given by displacement of cycles along closed paths. The image of this representation of $$\pi_ 1({\mathbb{C}}^{\mu}\setminus \Delta)$$ in $$GL_{\mu}({\mathbb{Z}})$$ is called the monodromy group of the versal family. The monodromy group can be thought of as a subgroup of the isotropy group of the intersection form on the intermediate homology $$H_{n- 1}(V_{\lambda},Z)$$. The classical Picard-Lefschetz theory describes the generators of the monodromy group of a versal deformation (of isolated hypersurface singularities) in terms of a particular basis in $$H_{n- 1}(V_{\lambda},{\mathbb{Z}})$$, the so-called basis of vanishing cycles.
In the present paper, the author extends the classical Picard-Lefschetz theory, in that he investigates the action of the fundamental group $$\pi_ 1({\mathbb{C}}^{\mu}\setminus \Delta)$$ on the homology groups $$H_ n({\mathbb{C}}^ n\setminus V_{\lambda},{\mathcal Z}(q))$$ of the complement of a hypersurface $$V_{\lambda}$$ with values in a certain non-trivial locally constant sheaf $${\mathcal Z}(q)$$. This “twisted” sheaf, with stalk $${\mathbb{Z}}[q,q^{-1}]$$, involves the complex variable q and is characterized by the property that a closed path around $$V_{\lambda}$$ induces the multiplication by q on its stalks. The main result of the paper provides an explicit set of generators of the monodromy group of the given versal deformation in $$H_ n({\mathbb{C}}^ n\setminus V_{\lambda},{\mathcal Z}(q))$$, and that in terms of a suitable homology basis. In the concluding section of his paper, the author discusses examples, applications, and possible generalizations of his approach to the monodromy group. He shows how the Burau representation of the braid group $$\pi_ 1({\mathbb{C}}^{\mu}\setminus \Delta)$$ can be obtained, how the twisted cohomology of the complement of a non-singular hypersurface V in $${\mathbb{C}}^ n$$ can be interpreted as the De Rham cohomology of a certain complex of singular differential forms on $${\mathbb{C}}^ n\setminus V$$, and how the classical monodromy operator and the signature of the intersection form are calculated from his twisted Picard-Lefschetz formulae.
Reviewer: W.Kleinert

##### MSC:
 32S30 Deformations of complex singularities; vanishing cycles 32Sxx Complex singularities 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
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