Roberto Giménez Conejero

We prove that, if two germs of plane curves \((C,0)\) and \((C',0)\) with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then \(C\) is complex isomorphic to \(C'\) or to \(\overline{C'}\). A similar result was shown by Ephraim for irreducible hypersurfaces before, but his proof is not constructive. Indeed, we show that the complex isomorphism is given by the Taylor series of the diffeomorphism. We also prove an analogous result for the case of non-irreducible hypersurfaces containing an irreducible component of zero-dimensional isosingular locus. Moreover, we provide a general overview of the different classifications of plane curve singularities.

We prove that a map germ \(f:(\mathbb{C}^n,S)\to(\mathbb{C}^{n+1},0)\) with isolated instability is stable if and only if \(\mu_I(f)=0\), where \(\mu_I(f)\) is the image Milnor number defined by Mond. In a previous paper we proved this result with the additional assumption that \(f\) has corank one. The proof here is also valid for corank \(\ge 2\), provided that \((n,n+1)\) are nice dimensions in Mather's sense (so \(\mu_I(f)\) is well defined). Our result can be seen as a weak version of a conjecture by Mond, which says that the \(\mathscr{A}_e\)-codimension of \(f\) is \(\le \mu_I(f)\), with equality if \(f\) is weighted homogeneous. As an application, we deduce that the bifurcation set of a versal unfolding of \(f\) is a hypersurface.

We give the definition of the Thom condition and we show that given any germ of complex analytic function \(f\colon(X,x)\to(\mathbb{C},0)\) on a complex analytic space \(X\), there exists a geometric local monodromy without fixed points, provided that \(f\in\mathfrak m_{X,x}^2\), where \(\mathfrak m_{X,x}\) is the maximal ideal of \(\mathcal{O}_{X,x}\). This result generalizes a well-known theorem of the second named author when \(X\) is smooth and proves a statement by Tibar in his PhD thesis. It also implies the A'Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that \(X\) has maximal rectified homotopical depth at \(x\) and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities.

We study germs of analytic maps \(f:(X,S)\to(\mathcal{C}^p,0)\), when \(X\) is an ICIS of dimension \(n< p\). We define an image Milnor number, generalizing Mond's definition, \(\mu_I(X,f)\) and give results known for the smooth case such as the conservation of this quantity by deformations. We also use this to characterise the Whitney equisingularity of families of corank one map germs \(f_t\colon(\mathbb{C}^n,S)\to(\mathbb{C}^{n+1},0)\) with isolated instabilities in terms of the constancy of the \(\mu_I^*\)-sequences of \(f_t\) and the projections \(\pi\colon D^2(f_t)\to\mathbb{C}^n\), where \(D^2(f_t)\) is the ICIS given by double point space of \(f_t\) in \(\mathbb{C}^n\times\mathbb{C}^n\). The \(\mu_I^*\)-sequence of a map germ consist of the image Milnor number of the map germ and all its successive transverse slices.

We show three basic properties on the image Milnor number \(\mu_I(f)\) of a germ \(f\colon(\mathbb{C}^n,S)\rightarrow(\mathbb{C}^{n+1},0)\) with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak version of the Mond conjecture, which says that \(\mu_I(f)=0\) if and only if \(f\) is stable. Finally, we show a conjecture by Houston that any family \(f_t\colon(\mathbb{C}^n,S)\rightarrow(\mathbb{C}^{n+1},0)\) with \(\mu_I(f_t)\) constant is excellent in Gaffney's sense. By technical reasons, in the two last properties we consider only the corank 1 case.

We compute the signature of the Milnor fiber of certain type of non-isolated complex surface singularities, namely, images of finitely determined holomorphic germs. An explicit formula is given in algebraic terms. As a corollary we show that the signature of the Milnor fiber is a topological invariant for these singularities. The proof combines complex analytic and smooth topological techniques. The main tools are Thom-Mather theory of map germs and the Ekholm-Szűcs-Takase-Saeki formula for immersions. We give a table with many examples for which the signature is computed using our formula.

We give a simple way to study the isotypical components of the homology of simplicial complexes with actions of finite groups, and use it for Milnor fibers of ICIS. We study the homology of images of mappings \(f_t\) that arise as deformations of complex map germs \(f:(\mathbb{C}^n,S)\to(\mathbb{C}^p,0)\), with \(nn+1\): stable perturbations with contractible image and homology of \(\text{im} f_t\) in unexpected dimensions. We show that Houston's conjecture, \(\mu_I\) constant in a family implies excellency in Gaffney's sense, is false, but we give a proof for the cases where it holds.