Inner geometry of complex surfaces: a valuative approach

Abstract

Given a complex analytic germ $(X,0)$ in $({ℂ}^n,0)$, the standard Hermitian metric of ${ℂ}^n$ 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 nonarchimedean 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.