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

Huang, Xiaojun; Luk, Hing Sun; Yau, Stephen S.-T.

J. Math. Soc. Japan 55, No. 3, 633-640 (2003).

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.

Reviewer: Gerhard Pfister (Kaiserslautern)

MSC:

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

Keywords:

de Rham cohomology; Milnor number; Poincaré number; isolated hypersurface singularity; Tjurina number