The main goal of the present article is the computation of the Heegaard Floer homology introduced by Ozsváth and Szabó for a family of plumbed rational homology 3-spheres. The main motivation is the study of the Seiberg-Witten type invariants of links of normal surface singularities.

We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we establish its validity for a large class of singularities: some rational and minimally elliptic (including the cyclic quotient and "polygonal") singularities, and Brieskorn-Hamm complete intersections. Some of the verifications are based on a result which describes (in terms of the plumbing graph) the Reidemeister-Turaev sign refined torsion (or, equivalently, the SeibergWitten invariant) of a rational homology 3-manifold M , provided that M is given by a negative definite plumbing.These results extend previous work of Artin, Laufer and S S-T Yau, respectively of Fintushel-Stern and Neumann-Wahl.

For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U ]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of Ozsváth and Szabó, but it has even more structure. If M is a complex singularity link then the normalized Euler-characteristic can be compared with the analytic invariants. The Seiberg-Witten Invariant Conjecture of [16], [13] is discussed in the light of this new object. §1. IntroductionThe article is a symbiosis of singularity theory and low-dimensional topology. Accordingly, it is preferable to separate its goals in two categories.From the point of view of 3-dimensional topology, the article contains the following main result. For every negative definite plumbed 3-manifold it constructs a graded Z[U ]-module from the combinatorics of the plumbing graph. This for rational homology spheres conjecturally equals the Heegaard-Floer homology of Ozsváth and Szabó. In fact, it has more structure (e.g. instead of a

In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

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.

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