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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

##### MSC:
 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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