We say that a contact manifold (M, ξ) is Milnor fillable if it is contactomorphic to the contact boundary of an isolated complex-analytic singularity (X , x). In this article we prove that any 3-dimensional oriented manifold admits at most one Milnor fillable contact structure up to contactomorphism. The proof is based on Milnor open books: we associate with any holomorphic function f : (X , x) → (C, 0), with isolated singularity at x (and any euclidian rug function ρ), an open book decomposition of M , and we verify that all these open books carry the contact structure ξ of (M, ξ) -generalizing results of Milnor and Giroux.

We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing structure on the abstract boundaries (also called links) of normal surface singularities. The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by Hirzebruch between the continued fraction expansions of two numbers λ > 1 and λ λ−1 .

The oriented link of the cyclic quotient singularity 𝒳p, q is orientation‐preserving diffeomorphic to the lens space L(p, q) and carries the standard contact structure ξst. Lisca classified the Stein fillings of (L(p, q), ξst) up to diffeomorphisms and conjectured that they correspond bijectively through an explicit map to the Milnor fibres associated with the irreducible components (all of them being smoothing components) of the reduced miniversal space of deformations of 𝒳p, q. We prove this conjecture using the smoothing equations given by Christophersen and Stevens. Moreover, based on a different description of the Milnor fibres given by de Jong and van Straten, we also canonically identify these fibres with Lisca's fillings. Using these and a newly introduced additional structure (the order) associated with lens spaces, we prove that the above Milnor fibres are pairwise non‐diffeomorphic (by diffeomorphisms which preserve the orientation and order). This also implies that de Jong and van Straten parametrize in the same way the components of the reduced miniversal space of deformations as Christophersen and Stevens.

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This text is a greatly expanded version of the mini-course I gave during the school Winter Braids VI organized in Lille between 22-25 February 2016. It is an introduction to the study of interactions between singularity theory of complex analytic varieties and contact topology. I concentrate on the relation between the smoothings of singularities and the Stein fillings of their contact boundaries. I tried to explain basic intuitions and facts in both fields, for the sake of the readers who are not accustomed with one of them.

Given an integral domain A, a monic polynomial P of degree n with coe cients in A and a divisor d of n, invertible in A, there is a unique monic polynomial Q such that the degree of P ? Q d is minimal for varying Q. This Q, whose d-th power best approximates P, is called the d-th approximate root of P. If f 2 C X]] Y ] is irreducible, there is a sequence of characteristic approximate roots of f, whose orders are given by the singularity structure of f. This sequence gives important information about this singularity structure. We study its properties in this spirit and we show that most of them hold for the more general concept of semiroot. We show then how this local study adapts to give a proof of Abhyankar-Moh's embedding line theorem.

Abstract. -Let (S, 0) be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor E and its irreducible components E i , i ∈ I. The Nash map associates to each irreducible component C k of the space of arcs through 0 on S the unique component of E cut by the strict transform of the generic arc in C k . Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if E · E i < 0 for any i ∈ I.Résumé (Une classe de singularités non-rationnelles de surfaces ayant une application de Nash bijective) Soit (S, 0) un germe de surface analytique complexe normale. Nous considérons le diviseur exceptionnel réduit E et ses composantes irréductibles E i , i ∈ I sur sa réso-lution minimale. L'application de Nash associeà chaque composante irréductible C k de l'espace des arcs passant par 0 sur S, l'unique composante de E rencontrée par la transformée stricte de l'arc générique dans C k . Nash a prouvé son injectivité et a demandé si elleétait bijective. Nous prouvons que c'est le cas si E · E i < 0 pour tout i ∈ I comme cas particulier de notre théorème principal.

We associate to any irreducible germ S of complex quasi-ordinary hypersurface an analytically invariant semigroup. We deduce a direct proof (without passing through their embedded topological invariance) of the analytical invariance of the normalized characteristic exponents. These exponents generalize the generic Newton-Puiseux exponents of plane curves. Incidentally, we give a toric description of the normalization morphism of the germ S.