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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

Given a complex analytic germ (X,0) in (n,0), the standard Hermitian metric of 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.

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
Received: 24 September 2019
Revised: 12 October 2020
Accepted: 16 February 2021
Published: 5 April 2022
Proposed: Lothar Göttsche
Seconded: Mark Gross, Walter Neumann
André Belotto da Silva
Institut de Mathématiques de Marseille
Aix-Marseille Université
Lorenzo Fantini
Centre de Mathématiques Laurent Schwartz
École Polytechnique
Anne Pichon
Institut de Mathématiques de Marseille
Aix-Marseille Université