Inner geometry of complex surfaces and Laplacian formula

Given a complex analytic germ $(X, 0) \subset ( C^n, 0)$, 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 a family of natural numerical invariants, called inner rates.  I will explain how these inner rates can be computed from the 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 $(X,0)$. Then I will explain some applications of this formula. In particular, I will show that this gives a first approach to the question of Lê Dung Tràng on the existence of a duality between the two algorithm of resolution of surface singularities, on one hand by a sequence of normalised blow-ups of points, and on the other hand by a sequence of normalised Nash transforms. This it a join work with André Belotto Da Silva and Lorenzo Fantini.