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The geometric genus of hypersurface singularities. (English) Zbl 1343.32020

Let \((X,0)\) be a normal surface singularity whose link \(M\) is a rational homology sphere. A path is a sequence of integral cycles supported on the exceptional curve of a fixed resolution, at each step increasing only by a base element, and connecting the trivial cycle to the anticanonical cycle. For such a path \(\gamma\), a path lattice cohomology \(\mathbb{H}^0(\gamma)\) is defined as well as its normalized rank \(eu(\mathbb{H}^0(\gamma))\). It is proved that \(p_g=\underset{\gamma}{\min}\;eu (\mathbb{H}^0(\gamma))\) in the following cases:
(a)
for super isolated singularities;
(b)
for singularities with non-degenerate Newton principal part;
(c)
if \(\mathbb{H}^q(M)=0\) for \(q\geq 1\) and the singular germ satisfies the so-called SWIC Conjecture (in particular, for all weighted homogeneous and minimally elliptic singularities).

MSC:

32S05 Local complex singularities
32S25 Complex surface and hypersurface singularities
32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
14B05 Singularities in algebraic geometry
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