×

The higher residue pairings \(K_ F^{(k)}\) for a family of hypersurface singular points. (English) Zbl 0565.32005

Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 2, 441-463 (1983).
[For the entire collection see Zbl 0509.00008.]
E. Brieskorn [Manuscr. Math. 2, 103-161 (1970; Zbl 0186.261)] studied the relative De Rham complex \({\mathcal H}_ F^{(0)}\) (here in fact simply a module) on the parameter space S of an unfolding F of an isolated hypersurface singularity f. It is endowed with the (regular) Gauss-Manin connection. Thereby a given vector field \(\delta\) on S defines a filtration \(\pi_*{\mathcal H}_ F^{(-k)}\) on the image \(\pi_*{\mathcal H}_ F^{(0)}\) under a projection \(\pi: S\to T\) parallel to integral curves to \(\delta\), as well an \({\mathcal O}_ T\)-dual filtration \({\mathcal H}^{v(k)}\) on the \({\mathcal O}_ T\)-dual \({\mathcal H}^{v(0)}\) to \(\pi_*{\mathcal H}_ F^{(0)}\). Since one has a sort of strict compatibility of the filtration with the \({\mathcal O}_ T\)-duality, one can define a self duality on \(\pi_*{\mathcal H}_ F^{(0)}/\pi_*{\mathcal H}_ F^{(-k-1})\) which gives rise to the definition of the pairings \(K^{(k)}: \pi_*{\mathcal H}_ F^{(0)}\times \pi_*{\mathcal H}_ F^{(0)}\to {\mathcal O}_ T\) mentioned in the title. After describing this construction, the author lists some properties of \(K^{(k)}\), one of which being the identification of \(K^{(0)}\) with some natural relative (with respect to \(X\to T)\) resdiue map.
Reviewer: H.Esnault

MSC:

32A27 Residues for several complex variables
32C30 Integration on analytic sets and spaces, currents
14J17 Singularities of surfaces or higher-dimensional varieties
14F40 de Rham cohomology and algebraic geometry