Normal surface singularities admitting contracting automorphisms

Favre, Charles; Ruggiero, Matteo

doi:10.5802/afst.1425

Cited by 4 publications

(5 citation statements)

References 34 publications

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“…By [FR14, Theorem A], it follows that (X, 0) is weighted homogeneous. Finally the proof of [FR14,Corollary B] shows that S(f N ) is a principal elliptic fiber bundle of Kodaira dimension 0 or 1 so that we are in case (i) of our theorem. Case 2b.…”

Section: Complex Analytic Sandwiched Singularitiesmentioning

confidence: 77%

“…Then (X ′ , 0 ′ ) is a cyclic singularity and (X, 0) is a quotient singularity. Again the proof of [FR14,Corollary B] shows that S(f ) is a Hopf surface. If the group G acting on (X ′ , 0 ′ ) is trivial, then (X, 0) is a cyclic quotient singularity, and the surface is either a secondary Hopf or a primary Hopf.…”

Section: Complex Analytic Sandwiched Singularitiesmentioning

confidence: 99%

“…But f is fixing 0 ′ , hence K = {0 ′ }. In other words, f : (X ′ , 0 ′ ) → (X ′ , 0 ′ ) is a contracting automorphism in the terminology of [FR14].…”

Section: Complex Analytic Sandwiched Singularitiesmentioning

confidence: 99%

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Links of sandwiched surface singularities and self-similarity

2019

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We characterize sandwiched singularities in terms of their link in two different settings. We first prove that such singularities are precisely the normal surface singularities having self-similar non-archimedean links. We describe this self-similarity both in terms of Berkovich analytic geometry and of the combinatorics of weighted dual graphs. We then show that a complex surface singularity is sandwiched if and only if its complex link can be embedded in a Kato surface in such a way that its complement remains connected.

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Section: Complex Analytic Sandwiched Singularitiesmentioning

confidence: 77%

Section: Complex Analytic Sandwiched Singularitiesmentioning

confidence: 99%

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Links of sandwiched surface singularities and self-similarity

2019

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“…, x n ). Let φ ∈ X ,0 be any holomorphic function at 0 ∈ X , and σ : (X , 0) → (X , 0) any automorphism (see [52,31] for constructions of automorphisms over quasihomogeneous singularities). We may consider the germ f = φ ω σ, defined in coordinates by f (x) = φ ω 1 σ 1 (x), .…”

Section: Quasihomogeneous Singularitiesmentioning

confidence: 99%

Local dynamics of non-invertible maps near normal surface singularities

Gignac¹,

Ruggiero²

2017

Preprint

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We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs f : (X , x 0 ) → (X , x 0 ), where X is a complex surface having x 0 as a normal singularity. We prove that as long as x 0 is not a cusp singularity of X , then it is possible to find arbitrarily high modifications π: X π → (X , x 0 ) such that the dynamics of f (or more precisely of f N for N big enough) on X π is algebraically stable. This result is proved by understanding the dynamics induced by f on a space of valuations associated to X ; in fact, we are able to give a strong classification of all the possible dynamical behaviors of f on this valuation space. We also deduce a precise description of the behavior of the sequence of attraction rates for the iterates of f . Finally, we prove that in this setting the first dynamical degree is always a quadratic integer.

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“…The existence of such a dynamical data gives restrictions on the geometry of the singularity. Indeed, in dimension 2 it has been proved in [FR14] that a normal surface singularity (X, 0) admitting a contracting automorphism is quasi-homogeneous. Up to finite quotients, quasi-homogeneous surface singularities are obtained by contracting the zerosection of a negative degree line bundle over a curve E. The induced "singular Hopf manifold" associated to such dynamical data can be a Hopf surface, a Kodaira surface, or of Kodaira dimension 1 according to the genus of E (see [Kat78], [BHPVdV04, Section V.5]).…”

Section: Tropical Hopf Manifoldsmentioning

confidence: 99%

Tropical Hopf manifolds and contracting germs

Ruggiero¹,

Shaw

2016

manuscripta math.

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Abstract. Classical Hopf manifolds are compact complex manifolds whose universal covering is C d {0}. We investigate the tropical analogues of Hopf manifolds, and relate their geometry to tropical contracting germs. To do this we develop a procedure called monomialization which transforms non-degenerate tropical germs into morphisms, up to tropical modification. A link is provided between tropical Hopf manifolds and the analytification of Hopf manifolds over a non-archimedean field. We conclude by computing the tropical Picard group and (p, q)-homology groups.

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Normal surface singularities admitting contracting automorphisms

Abstract: We show that a complex normal surface singularity admitting a contracting automorphism is necessarily quasihomogeneous. We also describe the geometry of a compact complex surface arising as the orbit space of such a contracting automorphism.

Cited by 4 publications

References 34 publications

Links of sandwiched surface singularities and self-similarity

Links of sandwiched surface singularities and self-similarity

Local dynamics of non-invertible maps near normal surface singularities

Tropical Hopf manifolds and contracting germs

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