The semi-ring structure and the spectral pairs of sesqui-linear forms. (English) Zbl 0814.11022

Némethi, András

Algebra Colloq. 1, No. 1, 85-95 (1994).

The main object under consideration is the quadruple \((U;b,h,V)\), where \(U\) is an \(n\)-dimensional \(\mathbb{C}\)-vector space, \(b : U \to U^*\) is an \(\varepsilon\)-symmetric \(\mathbb{C}\)-linear endomorphism, \(h\) is a \(b\)- orthogonal automorphism of \(U\), and \(V : U^* \to U\) is a \(\mathbb{C}\)-linear endomorphism such that \(\overline{\theta^{-1} \circ V^*} = - \varepsilon V \circ \overline{h^*}\), and \(V \circ b = h - I\) (here \(\varepsilon = (-1)^ n\) and \(\theta : U \to U^{**}\) is the natural isomorphism). This object is called an \(\varepsilon\)-hermitian variation structure (HVS) recalling that its origin lies in Hodge theory where to each analytic germ \(f : (\mathbb{C}^ n,0) \to (\mathbb{C},0)\) with isolated singularity at the origin one can associate its Seifert form \(V_ f\) together with the intersection form \(b_ f\) and the monodromy operator \(h_ f\) [see J. Scherk and J. H. M. Steenbrink, Math. Ann. 271, 641-665 (1985; Zbl 0618.14002) and references therein].
Following J. Milnor [Invent. Math. 8, 83-97 (1969; Zbl 0177.052)] and W. D. Neumann [Enseign. Math. 31, 223-232 (1983; Zbl 0586.14023)], the author gives a list of indecomposable HVS (with respect to the additive law). Passing to the corresponding graded object, he studies its multiplicative structure (with respect to tensor product) and computes “the multiplicative table”.
The more delicate investigation of the arising graded semirings leads to their “Hodge numbers” and “spectral pairs” as in the original geometric situation.

Reviewer: B.Kunyavskii (Ramat Gan)

MSC:

11E39	Bilinear and Hermitian forms
14B05	Singularities in algebraic geometry
14D07	Variation of Hodge structures (algebro-geometric aspects)
32S55	Milnor fibration; relations with knot theory
32S05	Local complex singularities
32S35	Mixed Hodge theory of singular varieties (complex-analytic aspects)
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)

Keywords:

Hermitian forms; mixed Hoge theory; local singularities; Milnor fibration; Hodge numbers; spectral pairs; hermitian variation structure