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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.

##### MSC:
 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)
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