Euler-homogeneous singularities and logarithmic differential forms

✍ Scribed by A. G. Aleksandrov

• Publisher: Springer
• Year: 1986
• Tongue: English
• Weight: 604 KB
• Volume: 4
• Category: Article
• ISSN: 0232-704X
• DOI: 10.1007/bf00129909

✦ Synopsis

In his article 18) K. Saito, computing the GauB-Manin connection of the versal deformation of the single singularity A3, introduced the notion of differentiable forms on smooth complex manifolds with logarithmic poles along an irreducible hypersurface -the discriminant of the deformation.

The coefficents of the connection can be expressed using these differentiable forms. The further development of this theory (cf. C 19 1, 20 ]) and its applications (cf. e.g. 19 ,

[23 ]) showed the particular importance of those hypersurfaces for which the 0 5 -modules 2Q (log D) are locally free. These hypersurfaces are called "Saito divisors". One may expect that the singularities of Saito divisors, which are not isolated and not normal, have special properties. In the present paper we shall analyze one of these properties, i.e. the Cohen-Macauly property of the singular subvariety

1. LOGARITHMIC DIFFERENTIAL FORMS AND VECTOR FIELDS

Let S be a smooth m-dimensional complex manifold and DC S a divisor whose irreducible components have multiplicity 1.

To such a divisor coherent analytic sheafes 1 (log D), q 0, and DerS(log D) are associated in the following way, cf. C 19 ]: