The geometric genus and Seiberg-Witten invariant of Newton nondegenerate surface singularities

TL;DR

This paper proves the Seiberg-Witten invariant conjecture for Newton nondegenerate hypersurface surface singularities, providing explicit methods to compute the geometric genus from the link using combinatorial and resolution techniques.

Contribution

It establishes the conjecture for a broad class of singularities and introduces a new computational approach based on resolution graphs and Newton diagrams.

Findings

The geometric genus can be explicitly recovered from the link for Newton nondegenerate singularities.

The normalized Seiberg-Witten invariant coincides with the geometric genus for these singularities.

The paper provides explicit resolution and computation methods using Oka's algorithm and combinatorial analysis.

Abstract

Given a normal surface singularity (X,0), its link, M is a closed differentiable three dimensional manifold which carries much analytic information. It is an interesting question to ask whether, under suitable analytic and topological conditions, the geometric genus (or other analytic invariants) can be recovered from the link. The Casson invariant conjecture predicts that p_g can be identified using the Casson invariant in the case when (X,0) is a complete intersection and M has trivial first homology with integral coefficients. The Seiberg-Witten invariant conjecture predicts that the geometric genus of a Gorenstein singularity, whose link has trivial first homology with rational coefficients, can be calculated as a normalized Seiberg-Witten invariant of the link. The first conjecture is still open, but counterexamples have been found for the second one. We prove here the Seiberg-Witten invariant conjecture for hypersurface singularities given by a function with Newton nondegenerate principal part. We provide a theory of computation sequences and of the way they bound the geometric genus. Newton nondegenerate singularities can be resolved explicitly by Oka's algorithm, and we exploit the combinatorial interplay between the resolution graph and the Newton diagram to show that in each step of the computation sequence we construct, the given bound is sharp. Our method recovers the geometric genus of (X,0) explicitly from the link, assuming that (X,0) is indeed Newton nondegenerate with a rational homology sphere link. Assuming some additional information about the Newton diagram, we recover part of the spectrum, as well as the Poincar\'e series associated with the Newton filtration. Finally, we show that the normalized Seiberg-Witten invariant associated with the canonical spin^c structure on the link coincides with our identification of the geometric genus.