CEU Electronic Theses and Dissertations, 2016
| Author | Sigurðsson, Baldur |
|---|---|
| Title | The geometric genus and Seiberg--Witten invariant of Newton nondegenerate surface singularities |
| Summary | Given a normal surface singularity (X, 0), its link, M, a closed differentiable three dimensional manifold, carries much analytic information. For example, a germ of a normal space is regular if (and only if) its link is the three sphere S^3 (it is even sufficient to assume that π_1(M) = 1 ). The geometric genus, p_g, is an analytic invariant of (X, 0) which, in general, cannot be recovered from the link. However, whether p_g = 0 can be determined from the link. The same holds for the statement p_g = 1, assuming that (X, 0) is Gorenstein. 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 original statement identifies the signature of a Milnor fiber rather then p_g, but in this case, these are equivalent data). 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 how 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 constructed, 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. With 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. |
| Supervisor | Némethi, András |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2016/sigurdsson_baldur.pdf |
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