Remarks on Laufer's formula for the Milnor number, Rochlin's signature theorem and the analytic Euler characteristic of compact complex manifolds (Q1692608)

Isolated hypersurface singularities \((V,0)\) defined by germs of holomorphic mappings \(f:(\mathbb{C}^{n+1},0)\rightarrow (\mathbb{C},0)\) can be studied using resolution of singularities \(\pi :\tilde{V}\rightarrow V\) or by considering the way noncritical levels of \(f\) degenerate to the special fiber of \(V\). Laufer's formula establishes a bridge between these two viewpoints. It says that for \(n=3\), one has \(\mu +1=\chi (\tilde{V})+K^2+12\rho _g\), where \(\mu\) is the Milnor number, \(\chi\) is the Euler characteristic, \(K^2\) is the self-intersection number of the canonical class of the resolution, and \(\rho _g\) is the geometric genus. In the paper this formula is considered in relationship with Gorenstein surface singularities and Rochlin's signature theorem.