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

##### 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)