[Paper Review] The geometric genus and Seiberg-Witten invariant of Newton nondegenerate surface singularities

Baldur Sigurðsson|arXiv (Cornell University)|Jan 11, 2016

Geometric and Algebraic Topology3 references4 citations

TL;DR

This paper proves the Seiberg–Witten invariant conjecture for Newton nondegenerate surface singularities by constructing explicit computation sequences that bound and ultimately recover the geometric genus from the link's topology. Using Oka's algorithm and combinatorial properties of the Newton diagram and resolution graph, the authors show that the normalized Seiberg–Witten invariant of the link's canonical spin^c structure equals the geometric genus, and they recover part of the spectrum and Poincaré series from topological data.

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 Poincaré 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.

Motivation & Objective

• To resolve the long-standing conjecture that the geometric genus of a Gorenstein surface singularity can be recovered from its link's Seiberg–Witten invariant.
• To establish a constructive method for computing the geometric genus from topological invariants of the link, specifically for Newton nondegenerate singularities.
• To demonstrate that the normalized Seiberg–Witten invariant of the canonical spin^c structure on the link equals the geometric genus.
• To recover part of the spectrum and the Poincaré series of the Newton filtration from the link's topology.

Proposed method

• Constructs computation sequences based on the resolution graph and Newton diagram to bound the geometric genus at each step.
• Employs Oka's algorithm to resolve Newton nondegenerate singularities via explicit combinatorial data from the Newton diagram.
• Uses lattice point counting in dilated polygons and polyhedra to compute intersection numbers and zeta function coefficients.
• Applies path lattice cohomology and the theory of the topological zeta function to relate analytic invariants to topological data.
• Leverages the polynomial part and periodic constant of power series in one variable to extract spectral information.
• Proves that the normalized Seiberg–Witten invariant of the canonical spin^c structure on the link equals the geometric genus via case analysis on arm configurations.

Experimental results

Research questions

• RQ1 Can the geometric genus of a Newton nondegenerate surface singularity be computed purely from topological data of its link?
• RQ2 Does the normalized Seiberg–Witten invariant of the canonical spin^c structure on the link coincide with the geometric genus?
• RQ3 Can part of the spectrum and the Poincaré series of the Newton filtration be recovered from the link's topology?
• RQ4 Under what combinatorial conditions on the Newton diagram is the geometric genus bound sharp at each step of the computation sequence?

Key findings

• The geometric genus of a Newton nondegenerate surface singularity is explicitly recoverable from the link's topology when the link is a rational homology sphere.
• The normalized Seiberg–Witten invariant of the canonical spin^c structure on the link equals the geometric genus, confirming the Seiberg–Witten invariant conjecture in this case.
• The computation sequences constructed are sharp at each step, ensuring exact recovery of the geometric genus through combinatorial means.
• The spectrum up to degree zero and the Poincaré series of the Newton filtration are recoverable from the link's topology under additional Newton diagram conditions.
• For singularities with a central triangle in the Newton diagram, the Seiberg–Witten coefficients vanish under specific geometric constraints, confirming the vanishing of certain zeta function terms.
• The method establishes a complete combinatorial correspondence between the resolution graph, Newton diagram, and analytic invariants like $p_g$ and the spectrum.