16 research outputs found
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 topological types of blow-up complexity ,
where is the -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 is . 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 , 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 self-dual
topological types of blow-up complexity . 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 be a normal complex analytic surface singularity. We say that is
arborescent if the dual graph of any resolution of it is a tree. Whenever
are distinct branches on , we denote by their intersection
number in the sense of Mumford. If is a fixed branch, we define when and
otherwise. We generalize a theorem of P{\l}oski concerning smooth germs of
surfaces, by proving that whenever is arborescent, then is an
ultrametric on the set of branches of different from . We compute the
maximum of , which gives an analog of a theorem of Teissier. We show that
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 and are smooth. We generalize
also from smooth germs to arbitrary arborescent ones their valuative
interpretation of the dual trees of the resolutions of . 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
of complex analytic curves contained in a smooth complex analytic surface .
The embedded topological type of such a pair is usually defined to be
that of the oriented link obtained by intersecting with a sufficiently
small oriented Euclidean sphere centered at the point , defined once a
system of local coordinates was chosen on the germ . If one
works more generally over an arbitrary algebraically closed field of
characteristic zero, one speaks instead of the combinatorial type of .
One may define it by looking either at the Newton-Puiseux series associated to
relative to a generic local coordinate system , 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 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 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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