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

A previous conjecture is verified for any normal surface singularitywhich admits a good C^*-action. This result connects the Seiberg–Witteninvariant 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 ofthe 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–Turaevsign-refined torsion and the Seiberg–Witten invariant(associated with any spin^c structure) of a Seifert 3-manifoldwith negative orbifold Euler number and genus zero.     

On Rational Cuspidal Projective Plane Curves

In 2002, L. Nicolaescu and the fourth author formulated a verygeneral conjecture which relates the geometric genus of a Gorensteinsurface singularity with rational homology sphere link withthe Seiberg--Witten invariant (or one of its candidates) ofthe link. Recently, the last three authors found some counterexamplesusing superisolated singularities. The theory of superisolatedhypersurface singularities with rational homology sphere linkis equivalent with the theory of rational cuspidal projectiveplane curves. In the case when the corresponding curve has onlyone singular point one knows no counterexample. In fact, inthis case the above Seiberg--Witten conjecture led us to a veryinteresting and deep set of ‘compatibility properties’of these curves (generalising the Seiberg--Witten invariantconjecture, but sitting deeply in algebraic geometry) whichseems to generalise some other famous conjectures and propertiesas well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yauinequalities). Namely, we provide a set of ‘compatibilityconditions’ which conjecturally is satisfied by a localembedded topological type of a germ of plane curve singularityand an integer d if and only if the germ can be realized asthe unique singular point of a rational unicuspidal projectiveplane curve of degree d. The conjectured compatibility propertieshave a weaker version too, valid for any rational cuspidal curvewith more than one singular point. The goal of the present articleis to formulate these conjectured properties, and to verifythem in all the situations when the logarithmic Kodaira dimensionof the complement of the corresponding plane curves is strictlyless than 2. 2000 Mathematics Subject Classification 14B05,14J17, 32S25, 57M27, 57R57 (primary), 14E15, 32S45, 57M25 (secondary).     

Hypersurface Singularities with 2-Dimensional Critical Locus

Consider an analytic germ f:(C^m, 0)(C, 0) (m3) whose criticallocus is a 2-dimensional complete intersection with an isolatedsingularity (icis). We prove that the homotopy type of the Milnorfiber of f is a bouquet of spheres, provided that the extendedcodimension of the germ f is finite. This result generalizesthe cases when the dimension of the critical locus is zero [8],respectively one [12]. Notice that if the critical locus isnot an icis, then the Milnor fiber, in general, is not homotopicallyequivalent to a wedge of spheres. For example, the Milnor fiberof the germ f:(C⁴, 0)(C, 0), defined by f(x₁, x₂, x₃, x₄) =x₁x₂x₃x₄ has the homotopy type of S¹xS¹xS¹. On the other hand,the finiteness of the extended codimension seems to be the rightgeneralization of the isolated singularity condition; see forexample [9–12, 17, 18]. In the last few years different types of ‘bouquet theorems’have appeared. Some of them deal with germs f:(X, x)(C, 0) wheref defines an isolated singularity. In some cases, similarlyto the Milnor case [8], F has the homotopy type of a bouquetof (dim X–1)-spheres, for example when X is an icis [2],or X is a complete intersection [5]. Moreover, in [13] Siersmaproved that F has a bouquet decomposition FF₀Sⁿ...Sⁿ (whereF₀ is the complex link of (X, x)), provided that both (X, x)and f have an isolated singularity. Actually, Siersma conjecturedand Tibr proved [16] a more general bouquet theorem for thecase when (X, x) is a stratified space and f defines an isolatedsingularity (in the sense of the stratified spaces). In thiscase F_iF_i, where the F_i are repeated suspensions of complexlinks of strata of X. (If (X, x) has the ‘Milnor property’,then the result has been proved by Lê; for details see[6].) In our situation, the space-germ (X, x) is smooth, but f hasbig singular locus. Surprisingly, for dim Sing f^–1(0)2,the Milnor fiber is again a bouquet (actually, a bouquet ofspheres, maybe of different dimensions). This result is in thespirit of Siersma's paper [12], where dim Sing f^–1(0)= 1. In that case, there is only a rather small topologicalobstruction for the Milnor fiber to be homotopically equivalentto a bouquet of spheres (as explained in Corollary 2.4). Inthe present paper, we attack the dim Sing f^–1(0) = 2 case.In our investigation some results of Zaharia are crucial [17,18].     

The Seiberg-Witten invariant conjecture for splice-quotients

Splice-quotient singularities were introduced recently and studiedintensively by Neumann and Wahl. For such a singularity we provethat the geometric genus can be recovered from the topologyof the singularity, namely from the Seiberg–Witten invariant(associated with the canonical spin^c structure) of the link.This answers positively the conjecture formulated by the firstauthor and Nicolaescu.