Extendability of holomorphic differential forms near isolated hypersurface singularities. (English) Zbl 0584.32018

Let (V,0) be the germ of an analytic variety with an isolated singularity. A holomorphic p-form \(\omega\) on \(V-\{0\}\) is called of first kind if there is a resolution \(\pi: \tilde V\to V\) such that \(\pi^*(\omega)\) extends to a holomorphic form on all of \(\tilde V.\) For \(p\leq \dim V-2\) the authors show that every p-form on \(V-\{0\}\) is of first kind, thus extending a result of G. M. Greuel in Math. Ann. 250, 157-173 (1980; Zbl 0417.14003) in the case of hypersurface singularities. For \(p=\dim V\) resp. \(p=\dim V-1\) the length of the quotient of the space of all holomorphic p-forms on \(V-\{0\}\) by the space of forms of the first kind is called the geometric genus resp. the irregularity of the singularity (V,0). In the case of hypersurface singularities there invariants are computed in terms of the Gauß- Manin-system of a function defining (V,0).
Reviewer: H.Knörrer


32Sxx Complex singularities
32C36 Local cohomology of analytic spaces
14B05 Singularities in algebraic geometry
32S45 Modifications; resolution of singularities (complex-analytic aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
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[1] Ph. Du Bois, Complexe de de Rham filtré d’une variété singulière. Bull. Soc. Math. France109, 41–81 (1981). · Zbl 0465.14009
[2] P. Deligne, Théorie de Hodge II. Publ. Math. IHES40, 5–58 (1971). · Zbl 0219.14007
[3] G.-M. Greuel, Dualität in der lokalen Kohomologie isolierter Singularitäten. Math. Ann.250, 157–173 (1980). · Zbl 0417.14003
[4] Ph. A. Griffiths, On the periods of certain rational integrals I. Annals of Math.90, 460–495 (1969). · Zbl 0215.08103
[5] E. J. N. Looijenga andJ. H. M. Steenbrink, Milnor number and Tjurina number of complete intersections. Math. Ann.271, 121–124 (1985). · Zbl 0552.14003
[6] M. Merle andB. Teissier, Conditions d’adjonction, d’après Du Val. Séminaire sur les singularités des surfaces. Lecture Notes in Math. vol.777, pp. 230–245. Springer Verlag, Berlin etc. 1980.
[7] M. Saito, On the exponents and the geometric genus of an isolated hypersurface singularity. Proc. Symp. Pure Math. vol.40 Part 2 (1983), 653–662. · Zbl 0545.14031
[8] M. Saito, Exponents and Newton polyhedra of isolated hypersurface singularities. Preprint Grenoble 1983. · Zbl 0628.32038
[9] J. Scherk andJ. H. M. Steenbrink, On the mixed Hodge structure on the cohomology of the Milnor fibre. Math. Ann.271, 641–665 (1985). · Zbl 0618.14002
[10] J. H. M. Steenbrink, Mixed Hodge structures associated with isolated singularities. Proc. Symp. Pure Math. vol.40 Part 2 (1983), 513–536. · Zbl 0515.14003
[11] J. H. M. Steenbrink, Vanishing theorems on singular spaces. In Proceedings of the Luminy Conference on Differential Systems and Singularities, Astérisque130, 330–341 (1985). · Zbl 0582.32039
[12] A. N. Vaechenko, Asymptotic Hodge structure in the vanishing cohomology. Math. USSR Izvestija18, 469–512 (1982). · Zbl 0489.14003
[13] J. Wahl, A characterization of quasi-homogeneous Gorenstein surface singularities. Compos. Math.55, 269–288 (1985). · Zbl 0587.14024
[14] S. S.-T. Yau, On irregularity and geometric genus of isolated singularities. Proc. Symp. Pure Math. vol.40 Part 2 (1983), 653–662. · Zbl 0518.14002
[15] H. Hironaka, Introduction to the theory of infinitely near singular points. Memorias de Matematica del Institute ”Jorge Juan”, No. 28. Consejo Superior de Investigaciones Cientificas. Madrid 1974. · Zbl 0366.32007
[16] J. Lipman, Free derivation modules on algebraic varieties. Amer. J. Math.87 (1965), 874–898. · Zbl 0146.17301
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