A formula to calculate the invariant $J$ of a quasi-homogeneous map germ

On the topology of the transversal slice of a quasi-homogeneous map germ

We consider a corank $1$, finitely determined, quasi-homogeneous map germ $f$ from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We describe the embedded topological type of a generic hyperplane section of $f(\mathbb{C}^2)$, denoted by $γ_f$, in terms of the weights and degrees of $f$. As a consequence, a necessary condition for a corank $1$ finitely determined map germ $g:(\mathbb{C}^2,0)\rightarrow (\mathbb{C}^3,0)$ to be quasi-homogeneous is that the plane curve $γ_g$ has either two or three characteristic exponents. As an application of our main result, we also show that any one-parameter unfo…

复加权齐次完全交芽的扰动上的快速消失周

Fast vanishing cycles on perturbations of complex weighted-homogeneous complete intersection germs

该研究对加权齐次完全交的扰动芽，通过权重确定其快速周的同伦型等性质，得到大量快速周，应用于构造奇异利普希茨结构及判断内度量锥的障碍。

Real map germs and higher open books

We present a general criterion for the existence of open book structures defined by real map germs $(\bR^m, 0) \to (\bR^p, 0)$, where $m> p \ge 2$, with isolated critical point. We show that this is satisfied by weighted-homogeneous maps. We also derive sufficient conditions in case of map germs with isolated critical value.

Moduli spaces of a family of topologically non quasi-homogeneous functions

We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi homogeneous. Each element f in this class induces a germ of foliation (df = 0). Proceeding similarly to the homogeneous case and the quasi homogeneous case treated by Genzmer and Paul, we describe the local moduli space of the foliations in this class and give analytic normal forms. We prove also the uniqueness of these normal forms.

Multiplicity and degree as bi-Lipschitz invariants for complex sets

We study invariance of multiplicity of complex analytic germs and degree of complex affine sets under outer bi-Lipschitz transformations (outer bi-Lipschitz homeomorphims of germs in the first case and outer bi-Lipschitz homeomorphims at infinity in the second case). We prove that invariance of multiplicity in the local case is equivalent to invariance of degree in the global case. We prove invariance for curves and surfaces. In the way we prove invariance of the tangent cone and relative multiplicities at infinity under outer bi-Lipschitz homeomorphims at infinity, and that the abstract topol…

Image Milnor number and $\mathscr{A}_e$-codimension for maps between weighted homogeneous irreducible curves

Let $(X,0)\subset (\mathbb{C}^n,0)$ be an irreducible weighted homogeneous singularity curve and let $f:(X,0)\to(\mathbb{C}^2,0)$ be a map germ finite, one-to-one and weighted homogeneous with the same weights of $(X,0)$. We show that $\mathscr{A}_e$-$codim(X,f)=μ_I(f)$, where $\mathscr{A}_e$-$codim(X,f)$ is the $\mathscr{A}_e$-codimension, i.e., the minimum number of parameters in a versal deformation and $μ_I(f)$ is the image Milnor number, i.e., the number of vanishing cycles in the image of a stabilisation of $f$.