András Némethi

On the Ozsvath-Szabo invariant of negative definite plumbed 3-manifolds

The main goal of the present article is the computation of the Heegaard Floer homology introduced by Ozsvath and Szabo 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.

WEAKLY ELLIPTIC GORENSTEIN SINGULARITIES OF SURFACES

The main message of the paper is that for Gorenstein singularities, whose (real) link is rational homology sphere, the Artin--Laufer program can be continued. Here we give the complete answer in the case of elliptic singularities. The main result of the paper says that in the case of an elliptic Gorenstein singularity whose link is rational homology sphere, the geometric genus is a topological invariant. Actually, it is exactly the length of the elliptic sequence in the minimal resolution (or, equivalently, in S. S.-T. Yaus terminology: these singularities are maximally elliptic). In the paper we characterize the singularities with this property, and we compute their Hilbert-Samuel function from their resolution graph (generalizing some results of Laufer and Yau). The obstruction for a normal surface singularity to be maximally elliptic can be connected with the torsion part of some Picard groups, this is the new idea of the paper.

Seiberg–Witten Invariants and Surface Singularities. II: Singularities with Good C*-Action

A previous conjecture is verified for any normal surface singularity which admits a good C ∗ -action. This result connects the Seiberg–Witten invariant of the link (associated with a certain ‘canonical’ spin c structure) with the geometric genus of the singularity, provided that the link is a rational homology sphere. As an application, a topological interpretation is found of the generalized Batyrev stringy invariant (in the sense of Veys) associated with such a singularity. The result is partly based on the computation of the Reidemeister–Turaev sign-refined torsion and the Seiberg–Witten invariant (associated with any spin c structure) of a Seifert 3-manifold with negative orbifold Euler number and genus zero.

On the monodromy of complex polynomials

We show that the monodromy operator at infinity plus the decomposition of the homology given by the vanishing cycles completely determine the homology monodromy representation of any complex polynomial.

Semicontinuity of the spectrum at infinity

We prove that, for an analytic family of “weakly tame” regular functions on an affine manifold, the spectrum at infinity of each function of the family is semicontinuous in the sense of Varchenko.

On the Casson Invariant Conjecture of Neumann–Wahl

In the article we prove the Casson Invariant Conjecture of Neumann--Wahl for splice type surface singularities. Namely, for such an isolated complete intersection, whose link is an integral homology sphere, we show that the Casson invariant of the link is one-eighth the signature of the Milnor fiber.

GRADED ROOTS AND SINGULARITIES

2 Normal surface singularities. 2 2.1 The link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 The combinatorics of the link. . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 The topology of the link. The Heegaard Floer homology HF+(−M). . . . . 5 2.4 Some analytic invariants of the singularity. . . . . . . . . . . . . . . . . . . . 6 2.5 Rational singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.6 Weakly elliptic singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.7 Almost rational singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Classification of Rational Unicuspidal Projective Curves whose Singularities Have one Puiseux Pair

It is a very old and interesting open problem to characterize those collections of embedded topological types of local plane curve singularities which may appear as singularities of a projective plane curve C of degree d. The goal of the present article is to give a complete (topological) classification of those cases when C is rational and it has a unique singularity which is locally irreducible (i.e., C is unicuspidal) with one Puiseux pair.

ON RATIONAL CUSPIDAL PLANE CURVES, OPEN SURFACES AND LOCAL SINGULARITIES

Let C be an irreducible projective plane curve in the complex projective space P(2). The classification of such curves, up to the action of the automorphism group PGL(3, C) on P(2), is a very difficult open problem with many interesting connections. The main goal is to determine, for a given d, whether there exists a projective plane curve of degree d having a fixed number of singularities of given topological type. In this note we are mainly interested in the case when C is a rational curve. The aim of this article is to present some of the old conjectures and related problems, and to complete them with some results and new conjectures from the recent work of the authors.

THE LINK OF {f(x, y) + z n = 0} AND ZARISKI'S CONJECTURE

We consider suspension hypersurface singularities of type g=f(x,y)+z^n, where f is an irreducible plane curve singularity. For such germs, we prove that the link of g determines completely the Newton pairs of f and the integer n except for two pathological cases, which can be completely described. Even in the pathological cases, the link and the Milnor number of g determine uniquely the Newton pairs of f and n. In particular, for such g, we verify Zariskis conjecture about the multiplicity. The result also supports the following conjecture formulated in the paper. If the link of an isolated hypersurface singularity is a rational homology 3-sphere then it determines the embedded topological type, the equivariant Hodge numbers and the multiplicity of the singularity. The conjecture is verified for weighted homogeneous singularities too.