Complex surface singularities from the combinatorial point of view
In this paper the interplay between graph theory and the topology of isolated complex surface singularities is studied. To begin with it is well known that the dual graph \((\Gamma, w,g)\) of a good resolution \(\pi: \widetilde {V}\to V\) of a normal surface singularity \((V,p)\) determines \(\widetilde {V}\) up to diffeomorphism. Here the vertices of the graph \(\Gamma\) correspond to the irreducible components \(E_i\) of the exceptional fibre of \(\pi\), the edges to the intersections of these components, the coordinates \(w_i\) of the vector \(w\) of weights to the self-intersection numbers and the coordinates \(g_i\) of the vector \(g\) to the genera of the Riemann surfaces \(E_i\). Moreover the intersection matrix \(E= (E_i, E_j)\) is always negative definite. On the other hand the intersection matrix \(E= \text{adj} (\Gamma)+ \text{diag} (w)\) of an (abstract) weighted graph \((\Gamma, w)\) is negative definite for almost all negative weights \(w_i\). Especially an (abstract) double-weighted graph \((\Gamma, w,g)\), \(w\in \mathbb{Z}^n\), \(g\in \mathbb{N}^n\), without loops is the dual graph of a normal surface singularity if and only if its intersection matrix is negative definite. Finally, the authors exploit this interplay in the case of Gorenstein singularities. A normal surface singularity is said to be numerically Gorenstein if the tangent bundle of \(V-p\) is topologically trivial over \(\mathbb{C}\) which amounts to say that the canonical class \(K\in H^2 (V, \mathbb{Q})\) is an integral linear combination of the \(E_i\) satisfying the adjunction formula \(2g-2= w+EK\). For an (abstract) weighted graph \((\Gamma,w)\) with non-singular intersection matrix \(E\) then it is shown that the set of vectors \(g\) of genera such that the adjunction formula \(2g-2= 2+EK\) has an integral solution \(K\) is infinite and parametrized by the integral solutions \(X\) of the congruence \(EX\equiv w(2)\). Finally the graph manifold \(X(\Gamma)\) obtained by plumbing admits a spin-structure if and only if \(K\) is even. For even weight \(w\) this is realized for infinitely many genera \(g\). Moreover if \(K=0\), then all weights equal \(-2\), all genera are 0 and \(\Gamma\) is one of the classical Dynkin diagrams \(A_n\), \(D_n\), \(E_6\), \(E_7\), \(E_8\). Thus the authors obtain a new characterization of rational double points since \(K=0\) is equivalent to a spin(3)-structure on \(X(\Gamma)\).
scientific article; zbMATH DE number 823483
scientific article