Inner geometry of complex surfaces: a valuative approach

Given a complex analytic germ .X; 0/ in .Cⁿ; 0/, the standard Hermitian metric of Cⁿ 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.