We study deformations of holomorphic function germs f : (X, 0) → C where (X, 0) is an ICIS. We present conditions for these deformations to have constant Milnor number, Euler obstruction and Bruce-Roberts number.

We show that a family of isolated complete intersection singularities (ICIS) with constant total Milnor number has no coalescence of singularities. This extends a well known result of Gabrielov, Lazzeri and Lê for hypersurfaces. We use A'Campo's theorem to see that the Lefschetz number of the generic monodromy of the ICIS is zero when the ICIS is singular. We give a pair applications for families of functions on ICIS which extend also some known results for functions on a smooth variety.

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We show that a family of isolated complete intersection singularities (ICIS) with constant total Milnor number has no coalescence of singularities. This extends a well-known result of Gabriélov, Lazzeri and Lê for hypersurfaces. We use A’Campo’s theorem to see that the Lefschetz number of the generic monodromy of the ICIS is zero when the ICIS is singular. We give a pair applications for families of functions on ICIS which extend also some known results for functions on a smooth variety.

We study the equisingularity of a family of function germs {f t : (X t , 0) → (C, 0)}, where (X t , 0) are d-dimensional isolated determinantal singularities. We define the (d − 1)th polar multiplicity of the fibers X t ∩ f −1 t (0) and we show how the constancy of the polar multiplicities is related to the constancy of the Milnor number of f t and the Whitney equisingularity of the family.