# Inner geometry of complex surfaces: a valuative approach

## TL;DR

This paper introduces a valuative approach to analyze the inner metric structure of complex surface singularities, providing a formula for the Laplacian of inner rates that links topology, hyperplane sections, and polar curves.

## Contribution

It develops a novel formula for the Laplacian of the inner rate function on the valuation space, connecting topological and geometric data to the metric structure of complex surface germs.

## Key findings

- Inner rates are determined by topology, hyperplane sections, and polar curves.
- The Laplacian formula links valuation theory with metric geometry.
- Global data fully determine the local metric structure.

## Abstract

Given a complex analytic germ $(X, 0)$ in $(\mathbb C^n, 0)$, the standard Hermitian metric of $\mathbb C^n$ induces a natural arc-length metric on $(X, 0)$, called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity $(X,0)$ by means of an infinite family of numerical analytic invariants, called inner rates. Our main result is a formula for the Laplacian of the inner rate function on a space of valuations, the non-archimedean link of $(X,0)$. We deduce in particular that the global data consisting of the topology of $(X,0)$, together with the configuration of a generic hyperplane section and of the polar curve of a generic plane projection of $(X,0)$, completely determine all the inner rates on $(X,0)$, and hence the local metric structure of the germ. Several other applications of our formula are discussed in the paper.