Inner geometry of complex surfaces: a valuative approach

Belotto da Silva, André; Fantini, Lorenzo; Pichon, Anne

doi:10.2140/gt.2022.26.163

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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 $(X, 0)$ 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.

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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

References

Publication

Received: 24 September 2019

Revised: 12 October 2020

Accepted: 16 February 2021

Published: 5 April 2022

Proposed: Lothar Göttsche

Seconded: Mark Gross, Walter Neumann

Authors

André Belotto da Silva
Institut de Mathématiques de Marseille Aix-Marseille Université CNRS Marseille France
https://andrebelotto.com
Lorenzo Fantini
Centre de Mathématiques Laurent Schwartz École Polytechnique CNRS Palaiseau France
https://lorenzofantini.eu/
Anne Pichon
Institut de Mathématiques de Marseille Aix-Marseille Université CNRS Marseille France
http://iml.univ-mrs.fr/~pichon/

Volume 26, issue 1 (2022)