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Inner geometry of complex surfaces: a valuative approach | | André Belotto da Silva
; Lorenzo Fantini
; Anne Pichon
; | | Date: |
5 May 2019 | | Abstract: | Given a complex analytic germ $(X, 0)$ in $(mathbb C^n, 0)$, the standard
Hermitian metric of $mathbb C^n$ 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. Our main result is a formula
for the Laplacian of the inner rate function on a space of valuations, the
non-archimedean 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 in the paper. | | Source: | arXiv, 1905.1677 | | Services: | Forum | Review | PDF | Favorites | |
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