Roberto Giménez Conejero

University of Valencia | UV · Department of Geometry and Topology

We give the definition of the Thom condition and we show that given any germ of complex analytic function $$f:(X,x)\rightarrow ({\mathbb {C}},0)$$ f : ( X , x ) → ( C , 0 ) on a complex analytic space X , there exists a geometric local monodromy without fixed points, provided that $$f\in {\mathfrak {m}}_{X,x}^2$$ f ∈ m X , x 2 , where $${\mathfrak...

We study germs of analytic maps f:(X,S)→(Cp,0), when X is an icis of dimension n<p. We define an image Milnor number, generalizing Mond's definition, μI(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...

We give a simple way to study the isotypes of the homology of simplicial complexes with actions of finite groups, and use it for Milnor fibers of \textsc{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 $n<p$, and the behaviour of singularities (in...

El Seminari Predoc es una comunidad de práctica organizada por los estudiantes de doctorado vinculados a la Facultat de Ciències Matemàtiques de la Universitat de València que permite dar a conocer su trabajo a toda la comunidad educativa. Este seminario tiene el objetivo de ayudar a los participantes a perfilar y mejorar sus ponencias de carácter...

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...

We have been able to prove a characterization of the Whitney equisingularity for families of corank one germs from C n to C n+1 using a few invariants. To do so, we have proven Houston's conjecture on excellent unfoldings, proving also some fundamental results of the image Milnor number. We have also expanded the theory of map germs on icis, we hav...

We 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 double point space in $\mathbb{C}^n\times\mathbb...

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 se...

We show three basic properties of the image Milnor number µI(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 Mond’s...

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 Mo...