#### Volume 26, issue 1 (2022)

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Inner geometry of complex surfaces: a valuative approach

### André Belotto da Silva, Lorenzo Fantini and Anne Pichon

Geometry & Topology 26 (2022) 163–219
DOI: 10.2140/gt.2022.26.163
##### Abstract

Given a complex analytic germ $\left(X,0\right)$ in $\left({ℂ}^{n},0\right)$, the standard Hermitian metric of ${ℂ}^{n}$ induces a natural arc-length metric on $\left(X,0\right)$, called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity $\left(X,0\right)$ 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 nonarchimedean link of $\left(X,0\right)$. We deduce in particular that the global data consisting of the topology of $\left(X,0\right)$, together with the configuration of a generic hyperplane section and of the polar curve of a generic plane projection of $\left(X,0\right)$, completely determine all the inner rates on $\left(X,0\right)$, and hence the local metric structure of the germ. Several other applications of our formula are discussed.

##### Keywords
surface singularities, metric geometry, inner rates, Lipschitz geometry, valuations, Milnor fibers, Lê–Greuel–Teissier formula
##### Mathematical Subject Classification 2010
Primary: 32S25, 57M27
Secondary: 13A18, 14B05, 32S55