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Inner geometry of complex
surfaces: a valuative approach
André Belotto da Silva, Lorenzo Fantini and Anne Pichon |
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Geometry & Topology 26 (2022) 163–219
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DOI: 10.2140/gt.2022.26.163
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Abstract |
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Given a complex analytic germ in , the standard Hermitian metric of induces a natural arc-length metric on , called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity 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 . We deduce in particular that the global data consisting of the topology of , together with the configuration of a generic hyperplane section and of the polar curve of a generic plane projection of , completely determine all the inner rates on , and hence the local metric structure of the germ. Several other applications of our formula are discussed. |
Keywords
surface singularities, metric geometry, inner rates,
Lipschitz geometry, valuations, Milnor fibers,
Lê–Greuel–Teissier formula
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Mathematical Subject Classification 2010
Primary: 32S25, 57M27
Secondary: 13A18, 14B05, 32S55
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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
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Authors |
| André Belotto da Silva | |
| Institut de Mathématiques de
Marseille Aix-Marseille Université CNRS Marseille France |
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| https://andrebelotto.com | |
| Lorenzo Fantini | |
| Centre de Mathématiques Laurent
Schwartz École Polytechnique CNRS Palaiseau France |
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| https://lorenzofantini.eu/ | |
| Anne Pichon | |
| Institut de Mathématiques de
Marseille Aix-Marseille Université CNRS Marseille France |
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| http://iml.univ-mrs.fr/~pichon/ | |
