Abstract
In this article we extend Milnor’s fibration theorem to the case of functions of the form \({f_{\bar{g}}}\) with f, g holomorphic, defined on a complex analytic (possibly singular) germ (X, 0). We further refine this fibration theorem by looking not only at the link of \({\{f\bar{g} = 0\}}\) , but also at its multi-link structure, which is more subtle. We mostly focus on the case when X has complex dimension two. Our main result (Theorem 4.4) gives in this case the equivalence of the following three statements:
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(i)
The real analytic germ \({f_{\bar{g}}:(X, p) \rightarrow ({\mathbb R}^2, 0)}\) has 0 as an isolated critical value;
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(ii)
the multilink \({L_f \cup -L_g}\) is fibered; and
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(iii)
if \({\pi: \tilde{X}\rightarrow X}\) is a resolution of the holomorphic germ \({fg: (X, p) \rightarrow ({\mathbb C}, 0)}\) , then for each rupture vertex (j) of the decorated dual graph of π one has that the corresponding multiplicities of f, g satisfy: \({m^f_j \not = m^g_j}\) .
Moreover one has that if these conditions hold, then the Milnor-Lê fibration \({\Psi_{f\bar{g}}: {\mathcal L}_X\backslash (L_f \cup L_g) \rightarrow {\mathbb S}^1_\eta}\) of \({f_{\bar{g}}}\) is a fibration of the multilink \({L_f \cup -L_g}\) . We also give a combinatorial criterium to decide whether or not the multilink \({L_f \cup -L_g}\) is fibered. If the meromorphic germ f/g is semitame, then we show that the Milnor-Lê fibration given by \({\Psi_{f\bar{g}}}\) is equivalent to the usual Milnor fibration given by \({f_{\bar{g}}/|f\bar{g}|}\) . We finish this article by discussing several realization problems.
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Research partially supported by CONACYT and DGAPA-UNAM, Mexico, and by CNRS and ECOS, France.
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Pichon, A., Seade, J. Fibred multilinks and singularities \({f \overline g}\) . Math. Ann. 342, 487–514 (2008). https://doi.org/10.1007/s00208-008-0234-3
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DOI: https://doi.org/10.1007/s00208-008-0234-3