Journal of the Mathematical Society of Japan

Punctured local holomorphic de Rham cohomology

Xiaojun HUANG, Hing Sun LUK, and Stephen S.-T. YAU

Full-text: Open access


Let V be a complex analytic space and x be an isolated singular point of V. We define the q-th punctured local holomorphic de Rham cohomology Hhq(V,x) to be the direct limit of Hhq(U-{x}) where U runs over strongly pseudoconvex neighborhoods of x in V, and Hhq(U-{x}) is the holomorphic de Rahm cohomology of the complex manifold U-{x}. We prove that punctured local holomorphic de Rham cohomology is an important local invariant which can be used to tell when the singularity (V,x) is quasi-homogeneous. We also define and compute various Poincaré number p˜x(i) and p¯x(i) of isolated hypersurface singularity (V,x).

Article information

J. Math. Soc. Japan, Volume 55, Number 3 (2003), 633-640.

First available in Project Euclid: 3 October 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14B15: Local cohomology [See also 13D45, 32C36] 32S05: Local singularities [See also 14J17] 32S10: Invariants of analytic local rings 32S25: Surface and hypersurface singularities [See also 14J17]

Holomorphic de Rham cohomology Punctured local holomorphic de Rham cohomology Isolated hypersurface singularity Milnor number Poincar\'{e} number


HUANG, Xiaojun; Sun LUK, Hing; S.-T. YAU, Stephen. Punctured local holomorphic de Rham cohomology. J. Math. Soc. Japan 55 (2003), no. 3, 633--640. doi:10.2969/jmsj/1191418993.

Export citation