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Hodge numbers attached to a polynomial map. (English) Zbl 0944.32029
Consider a polynomial map \(f :{\mathbb C}^{n+1} \longrightarrow {\mathbb C}\). For \(r\) a big enough real number the map \(1/f : Z:= Z_r :={\mathbb C}^{n+1} \setminus f^{-1}(D_r) \rightarrow D^*_{1-r}\) is a locally trivial \(C^{\infty}\) fibration. The map \(1/f\) can be compactified, and a fiber over 0 can be added, so that we get a projective map \(p : X \rightarrow D_{1-r}\). Write \(Y = p^{-1}(0)\), and \(X \setminus Z = Y \cup \Delta\), where \(\Delta\) is the union of irreducible components of \(X \setminus Z\) not contained in \(Y\). Consider the universal covering \({\mathbb H}\) of \(D^*_{1-r}\) and let \(\widetilde{Z}:= Z \times_{D_{1/r}} {\mathbb H}\), which has the homotopy type of a generic fiber of \(f\). The cohomology groups of \(\widetilde{Z}\) with rational coefficients carry a limit mixed Hodge structure. The one on \(n\)–th cohomology group is called the limit MHS at infinity.
The authors study the equivariant Hodge numbers of this limit MHS in case that for all \(t \in {\mathbb C}\) which is not a critical value of \(f\), the closure of \({f = t}\) in projective space is non-singular.

32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14D07 Variation of Hodge structures (algebro-geometric aspects)
Full Text: DOI Numdam EuDML
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