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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

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
Full Text: DOI
[1] V. I. Arnol’d, ”Normal forms of functions near degenerate critical points, the Weyl groups Ak, Dk, Ek, and Lagrangian singularities,” Funkts. Anal. Prilozhen.,5, No. 4, 3–25 (1972).
[2] V. I. Arnol’d, ”Critical points of functions on manifolds with boundary, simple Lie groups Bk, Ck, F4, and singularities of evolutes,” Usp. Mat. Fiz.,33, No. 5, 91–105 (1978).
[3] V. I. Arnol’d, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps [in Russian], Vol. 1, Nauka, Moscow (1982).
[4] V. I. Arnol’d, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps [in Russian], Vol. 2, Nauka, Moscow (1984).
[5] I. N. Bernshtein, ”Analytic continuation of generalized functions with respect to a parameter,” Funkts. Anal. Prilozhen.,6, No. 4, 26–40 (1972).
[6] A. N. Varchenko, ”Semicontinuity of the spectrum and an upper bound for the number of singular points of a projective hypersurface,” Dokl. Akad. Nauk SSSR,270, No. 6, 1294–1297 (1983). · Zbl 0537.14003
[7] A. N. Varchenko, ”Local classification of volume forms in the presence of a hypersurface,” Funkts. Anal. Prilozhen.,19, No. 4, 23–31 (1985). · Zbl 0661.32017
[8] A. N. Varchenko, ”Combinatorics and topology of the disposition of affine hyperplanes in real space,” Funkts. Anal. Prilozhen.,21, No. 1, 11–22 (1987). · Zbl 0615.52005
[9] V. A. Vasil’ev, I. M. Gel’fand, and A. V. Zelevinskii, ”General hypergeometric functions on complex Grassmanians,” Funkts. Anal. Prilozhen.,21, No. 1, 23–38 (1987). · Zbl 0614.33008
[10] A. B. Givental’, ”Lagrangian manifolds with singularities and irreducible s2-modules,” Usp. Mat. Nauk,38, No. 6, 109–110 (1983). · Zbl 0535.58016
[11] A. B. Givental’ and V. V. Shekhtman, ”Monodromy groups and Hecke algebras,” Usp. Mat. Nauk,42, No. 4, 138 (1987).
[12] V. V. Goryunov, ”Geometry of bifurcation diagrams of simple projections to a line,” Funkts. Anal. Prilozhen.,15, No. 2, 1–8 (1981). · Zbl 0463.30010 · doi:10.1007/BF01082373
[13] P. Griffiths and J. Harris, Principles of Algebraic Geometry [Russian translation], Mir, Moscow (1982). · Zbl 0531.14002
[14] V. P. Kostov, ”Versal deformations of differential forms of degree \(\alpha\) on the line,” Funkts. Anal. Prilozhen.,18, No. 4, 81–82 (1984). · Zbl 0582.32041 · doi:10.1007/BF01076376
[15] S. K. Lando, ”Normal forms of powers of the volume form,” Funkts. Anal. Prilozhen.,19, No. 2, 78–79 (1985). · Zbl 0589.32048
[16] J. Milnor, Singular Points of Complex Hypersurfaces [Russian translation], Mir, Moscow (1971). · Zbl 0224.57014
[17] F. Fam, ”Generalized Picard–Lefschetz formulas and branched integrals,” Matematika,13, No. 4, 61–93 (1969).
[18] K. Aomoto, ”On the structure of integrals of power product of linear functions,” Sc. Papers Coll. General Educ. Univ. Tokyo,27, 49–61 (1977). · Zbl 0384.35045
[19] T. Kohno, ”Linear representations of braid groups and classical Yang–Baxter equation,” in: Artin’s Braid Groups, Santa Cruz (1987). · Zbl 0634.58040
[20] J. Steenbrink, ”Mixed Hodge structure on the vanishing cohomology,” in: Real and Complex Singularities, Nordic Summer School, Oslo (1976). · Zbl 0373.14007
[21] J. Steenbrink, ”Semicontinuity of the singularity spectrum,” Invent. Math.,79, No. 3, 557–565 (1985). · Zbl 0568.14021 · doi:10.1007/BF01388523
[22] W. Ebeling, ”On the monodromy groups of singularities,” in: Proc. Symp. Pure Math., Vol. 40, Part 1 (1983), pp. 327–336. · Zbl 0518.32007
[23] S. V. Chmutov, ”Monodromy groups of singularities of functions of two variables,” Funkts. Anal. Prilozhen.,15, No. 1, 61–66 (1981). · Zbl 0464.51002 · doi:10.1007/BF01082383
[24] G. G. Il’yuta, ”Monodromy and vanishing cycles of boundary singularities,” Funkts. Anal. Prilozhen.,19, No. 3, 11–21 (1985).
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