Lipschitz geometry of complex surface germs via inner rates of primary ideals
Yenni Cherik
TL;DR
This workstudies the metric (Lipschitz) geometry of normal complex surface germs by linking outer geometry to inner-rate data attached to $\mathfrak{m}$-primary ideals. It introduces the inner-rate function $\mathcal{I}_I$ on the non-archimedean link $\text{NL}(X,0)$, shows it is determined by resolution data and linear on dual-graph edges, and provides a geometric realization via a map $F$ that preserves the inner-rate metric. A key result is that distinct integrally closed $\mathfrak{m}$-primary ideals yield non-isomorphic inner-rate data, enabling the construction of infinite families of germs with the same normalization but distinct outer Lipschitz types. The paper also connects these inner-rate invariants to polar exploration and to the outer Lipschitz geometry via the triplet $(\Gamma_\pi,L_\pi,P_\pi)$, echoing and extending prior BFNP and NP2016 results. Practically, this yields explicit mechanisms to generate and distinguish non-equivalent outer Lipschitz types among germs sharing the same topological or analytic normalization.
Abstract
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.