arXiv subjects

Yenni Cherik

Publications and source records attributed to Yenni Cherik.

3 recordsLinked to original sources

arXiv2026

Inner Lipschitz Geometry of Complex Surface Germs with Non-Isolated Singularities: A Complete Classification

Let $(X,0)$ be a germ of a reduced and irreducible complex surface embedded in $(\mathbb{C}^k,0)$. In this paper, we give a complete invariant of the inner Lipschitz geometry of complex surface germs, extending the result of Birbrair--Neumann--Pichon \cite{BNP} to the non-isolated case. This invariant is expressed in terms of numerical invariants associated with the coordinate functions $f_1,\dots,f_k$ of the normalization map $n:(\overline{X},0)\to (X,0)\subset(\mathbb{C}^k,0)$, together with the combinatorics of a suitable good resolution of $(\overline{X},0)$.

math.AG↗

arXiv2025

Lipschitz geometry of complex surface germs via inner rates of primary ideals

Let $(X, 0)$ be a normal complex surface germ embedded in $(\mathbb{C}^n, 0)$, and denote by $\mathfrak{m}$ the maximal ideal of the local ring $\mathcal{O}_{X,0}$. In this paper, we associate to each $\mathfrak{m}$-primary ideal $I$ of $\mathcal{O}_{X,0}$ a continuous function $\mathcal{I}_I$ defined on the set of positive (suitably normalized) semivaluations of $\mathcal{O}_{X,0}$. We prove that the function $\mathcal{I}_{\mathfrak{m}}$ is determined by the outer Lipschitz geometry of the surface $(X, 0)$. We further demonstrate that for each $\mathfrak{m}$-primary ideal $I$, there exists a complex surface germ $(X_I, 0)$ with an isolated singularity whose normalization is isomorphic to $(X, 0)$ and $\mathcal{I}_I = \mathcal{I}_{\mathfrak{m}_I}$, where $\mathfrak{m}_I$ is the maximal ideal of $\mathcal{O}_{X_I,0}$. Subsequently, we construct an infinite family of complex surface germs with isolated singularities, whose normalizations are isomorphic to $(X,0)$ (in particular, they are homeomorphic to $(X,0)$) but have distinct outer Lipschitz types.

math.AG↗

arXiv2024

Inner rates of finite morphisms

Let $(X, 0)$ be a complex analytic surface germ embedded in $(\mathbb{C}^n,0)$ with an isolated singularity and $Φ=(g,f):(X,0) \longrightarrow (\mathbb{C}^2,0)$ be a finite morphism. We define a family of analytic invariants of the morphism $Φ$, called inner rates of $Φ$. By means of the inner rates we study the polar curve associated to the morphism $Φ$ when fixing the topological data of the curve $(gf)^{-1}(0)$ and the surface germ $(X,0)$, allowing to address a problem called polar exploration. We also use the inner rates to study the geometry of the Milnor fibers of a non constant holomorphic function $f:(X,0) \longrightarrow (\mathbb{C},0)$. The main result is a formula which involves the inner rates and the polar curve alongside topological invariants of the surface germ $(X,0)$ and the curve $(gf)^{-1}(0)$.

math.AG↗