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On the Milnor fibration for \(f({\mathbf {z}})\,{\overline{g}}({\mathbf {z}})\)

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Abstract

We consider a mixed function of type \(H({\mathbf {z}},\overline{{\mathbf {z}}})=f({\mathbf {z}})\,{\overline{g}}({{\mathbf {z}}})\) where f and g are convenient holomorphic functions which have isolated critical points at the origin, and assume that the intersection \(f=g=0\) is a complete intersection variety with an isolated singularity at the origin and H satisfies the multiplicity condition. We show that H has a tubular Milnor fibration at the origin. We also prove that H has a spherical Milnor fibration, provided the Newton non-degeneracy of the intersection variety \(f=g=0\) and Newton multiplicity condition hold. Finally, we give examples of f, g such that H does not satisfy the Newton multiplicity condition and it has or has not Milnor fibration.

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Acknowledgements

We thank the referee for indicating papers related to results presented here.

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  1. Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan

    Mutsuo Oka

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Oka, M. On the Milnor fibration for \(f({\mathbf {z}})\,{\overline{g}}({\mathbf {z}})\). European Journal of Mathematics 6, 998–1019 (2020). https://doi.org/10.1007/s40879-019-00380-1

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