The geometric genus of hypersurface singularities (Q277532)

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: {\parindent=0.6cm\begin{itemize} \item[(a)] for super isolated singularities; \item [(b)] for singularities with non-degenerate Newton principal part; \item [(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). NEWLINENEWLINE\end{itemize}}