Seiberg–Witten invariants and surface singularities

Abstract

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.

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

“…For plumbed 3-manifold M (as in Section 2.1), [12,Section 5.7] provides a combinatorial formula for T M, [k] (1) in terms of Γ (involving regularized FourierDedekind sums).…”

“…For plumbed manifolds M , it has a combinatorial formula from Γ (provided by A. Raţiu in his dissertation; and it can be deduced from the surgery formulas of [7] as well; see also [12,Section 5.3]):…”

“…The rational number K 2 + s does not depend on the (negative definite) plumbing graph, it is an invariant of M . It can be computed from Γ as follows (see [12,Section 5.2]): …”

“…For the extension of the conjecture [12] to arbitrary spin c structures, see [11]. For different definitions regarding surface singularities, the reader is invited to consult [10].…”

“…For Gorenstein weakly elliptic singularities, the equality p g = l follows from [9]. For a generalization of the conjecture [12] and the construction of those line bundles L for which one expects equality above, see [11].…”