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Irregular fibers of complex polynomials in two variables. (English) Zbl 1060.32015
Let $$f:\mathbb C^2 \rightarrow \mathbb C$$ be a polynomial and $$\mathcal B \subset \mathbb C$$ the bifurcation set. Let $$F_c=f^{-1}(c)$$. The fiber $$F_c$$ is called irregular if $$c \in \mathcal B$$. Let $$D_{\varepsilon}(c) \subset \mathbb C$$ be the disc of radius $$\varepsilon$$ centered at $$c$$ and $$T_c=f^{-1}(D_{\varepsilon}(c))$$. The value $$c \in \mathbb C$$ is called regular at infinity if there exist a compact set $$K$$ in $$\mathbb C^2$$ such that the restriction of $$f:T_c \setminus K \rightarrow D_{\varepsilon}(c)$$ is locally trivial.
The morphism $$j_c:H_1(F_c) \rightarrow H_1(T_c)$$ induced by the inclusion is studied. It is proved that it is an isomorphism if and only if $$c$$ is regular at infinity. Necessary and sufficient conditions for this morphism to be injective respectively surjective are given. The results are applied to the study of vanishing and invariant cycles.
##### MSC:
 32S20 Global theory of complex singularities; cohomological properties 32S45 Modifications; resolution of singularities (complex-analytic aspects) 32S30 Deformations of complex singularities; vanishing cycles
##### Keywords:
irregular fibers; vanishing cycles; invariant cycles
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