Period Mapping Associated to a Primitive Form

Abstract

The aim of this note is to give a summary of the study of primitive forms and period mappings associated with a universal unfolding F of an isolated hypersurface singularity, which was studied in [35]. There is already a summary [36] on the subject. Compared to that, this note contains mainly two new topics, covering some recent developments in the subject. The details will appear elsewhere. First, in Section 5 of this note we give a description of the construction of period mappings using a sequence of Ss-modules ^? ( F keZ instead of ^(0)modules J>^F keZ. This construction might give another insight into the relationship between Poincare duality of the Milnor fiber and the period mapping of the family F. This part is based on the lectures by the author at R. T. M. S., Kyoto University in the springs '81 and '82. Secondly, in Section 4 of this note, the existence of primitive forms is reduced to the existence of certain good sections of the principal symbol module q*Qp i^° the <f(0)-module ^FThe technique of the proof comes from the solution of Riemann-Hilbert problem on P by B. Malgrange [23] [24] [25] and Birkhoff [4]. The author is grateful to Professor Malgrange for helpful discussions at the Institute Fourier in Grenoble, March '83. Combining this result with a recent result by M. Saito [46], we are now able to construct primitive forms for a large class of singularities. For the moment the period mapping associated to a primitive form has been studied explicitely only for the cases of simple singularities and simple elliptic singularities (cf. [33] [42] [43]) which might give another approach to studying universal family of such singularities by E. Brieskorn [6] [7],