Milnor open books and Milnor fillable contact 3-manifolds

We say that a contact manifold is Milnor fillable if it is contactomorphic to the contact boundary of an isolated complex-analytic singularity (X,x). Generalizing results of Milnor and Giroux, we associate to each holomorphic function f defined on X, with isolated singularity at x, an open book which supports the contact structure. Moreover, we prove that any 3-dimensional oriented manifold admits at most one Milnor fillable contact structure up to contactomorphism. * * * * * * * * In the first version of the paper, we showed that the open book associated to f carries the contact structure only up to an isotopy. Here we drop this restriction. Following a suggestion of Janos Kollar, we also give a simplified proof of the algebro-geometrical theorem 4.1, central for the uniqueness result.Comment: 17 page

Fibonacci numbers and self-dual lattice structures for plane branches

Consider a plane branch, that is, an irreducible germ of curve on a smooth complex analytic surface. We define its blow-up complexity as the number of blow-ups of points necessary to achieve its minimal embedded resolution. We show that there are $F_{2n-4}$ topological types of blow-up complexity $n$, where $F_{n}$ is the $n$-th Fibonacci number. We introduce complexity-preserving operations on topological types which increase the multiplicity and we deduce that the maximal multiplicity for a plane branch of blow-up complexity $n$ is $F_n$. It is achieved by exactly two topological types, one of them being distinguished as the only type which maximizes the Milnor number. We show moreover that there exists a natural partial order relation on the set of topological types of plane branches of blow-up complexity $n$, making this set a distributive lattice, that is, any two of its elements admit an infimum and a supremum, each one of these operations beeing distributive relative to the second one. We prove that this lattice admits a unique order-inverting bijection. As this bijection is involutive, it defines a duality for topological types of plane branches. The type which maximizes the Milnor number is also the maximal element of this lattice and its dual is the unique type with minimal Milnor number. There are $F_{n-2}$ self-dual topological types of blow-up complexity $n$. Our proofs are done by encoding the topological types by the associated Enriques diagrams.Comment: 21 pages, 16 page

Ultrametric spaces of branches on arborescent singularities

Let $S$ be a normal complex analytic surface singularity. We say that $S$ is arborescent if the dual graph of any resolution of it is a tree. Whenever $A,B$ are distinct branches on $S$, we denote by $A \cdot B$ their intersection number in the sense of Mumford. If $L$ is a fixed branch, we define $U_L(A,B)= (L \cdot A)(L \cdot B)(A \cdot B)^{-1}$ when $A \neq B$ and $U_L(A,A) =0$ otherwise. We generalize a theorem of P{\l}oski concerning smooth germs of surfaces, by proving that whenever $S$ is arborescent, then $U_L$ is an ultrametric on the set of branches of $S$ different from $L$. We compute the maximum of $U_L$, which gives an analog of a theorem of Teissier. We show that $U_L$ encodes topological information about the structure of the embedded resolutions of any finite set of branches. This generalizes a theorem of Favre and Jonsson concerning the case when both $S$ and $L$ are smooth. We generalize also from smooth germs to arbitrary arborescent ones their valuative interpretation of the dual trees of the resolutions of $S$. Our proofs are based in an essential way on a determinantal identity of Eisenbud and Neumann.Comment: 37 pages, 16 figures. Compared to the first version on Arxiv, il has a new section 4.3, accompanied by 2 new figures. Several passages were clarified and the typos discovered in the meantime were correcte

The combinatorics of plane curve singularities. How Newton polygons blossom into lotuses

This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs $(C,o)$ of complex analytic curves contained in a smooth complex analytic surface $S$. The embedded topological type of such a pair $(S, C)$ is usually defined to be that of the oriented link obtained by intersecting $C$ with a sufficiently small oriented Euclidean sphere centered at the point $o$, defined once a system of local coordinates $(x,y)$ was chosen on the germ $(S,o)$. If one works more generally over an arbitrary algebraically closed field of characteristic zero, one speaks instead of the combinatorial type of $(S, C)$. One may define it by looking either at the Newton-Puiseux series associated to $C$ relative to a generic local coordinate system $(x,y)$, or at the set of infinitely near points which have to be blown up in order to get the minimal embedded resolution of the germ $(C,o)$ or, thirdly, at the preimage of this germ by the resolution. Each point of view leads to a different encoding of the combinatorial type by a decorated tree: an Eggers-Wall tree, an Enriques diagram, or a weighted dual graph. The three trees contain the same information, which in the complex setting is equivalent to the knowledge of the embedded topological type. There are known algorithms for transforming one tree into another. In this paper we explain how a special type of two-dimensional simplicial complex called a lotus allows to think geometrically about the relations between the three types of trees. Namely, all of them embed in a natural lotus, their numerical decorations appearing as invariants of it. This lotus is constructed from the finite set of Newton polygons created during any process of resolution of $(C,o)$ by successive toric modifications.Comment: 104 pages, 58 figures. Compared to the previous version, section 2 is new. The historical information, contained before in subsection 6.2, is distributed now throughout the paper in the subsections called "Historical comments''. More details are also added at various places of the paper. To appear in the Handbook of Geometry and Topology of Singularities I, Springer, 202

The linen industry of North-West England, 1660-1830

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