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Punctured local holomorphic de Rham cohomology. (English) Zbl 1034.32018

Let \((V,0) \subset (\mathbb{C}^{n+1},0)\) be the germ of an isolated hypersurface singularity. The \(q\)th punctured local holomorphic de Rham cohomology \(H^q_h(V,0)\) is defined as the direct limit of \(H^q_h (U \smallsetminus\{0\})\), where \(U\) runs over strongly pseudo convex neighbourhoods of \(0\) in \(V\) (here \(H^q_h (U \smallsetminus\{0\}\)) is the \(q\)th holomorphic de Rham cohomology of \(U \smallsetminus \{0\}\)).
It is proved that \[ \dim H^q_h(V,0) = 0 \] for \[ q \leq n - 2, \]
\[ \dim H^n_h(V,0) - \dim H^{n-1}_h (V,0) = \mu-\tau. \] Here \(\mu\) is the Milnor number and \(\tau\) the Tjurina number.

MSC:

32S05 Local complex singularities
14B05 Singularities in algebraic geometry
14B15 Local cohomology and algebraic geometry
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