The classification of taut surface singularities via representations from logarithmic vector fields

We study logarithmic vector fields on exceptional cycles through finite-dimensional representations and recover Laufer’s classification of taut normal complex surface singularities. For graphs with two nodes, restriction maps define representations of a fixed quiver, with dimensions bounded independently of the number of components and their self-intersections. We identify restriction cohomology with an extension group and use interval decomposition to reduce its vanishing to interpolation on a rational chain. These calculations determine the classification conditions, and vanishing on every analytic realization proves sufficiency, including the nonrational cases. The obstruction cycles also give a uniform criterion: a graph is taut if and only if $H^1(5E,S_X{\mathrm{\mid }}_{5E})=0$ for every analytic realization, where $S_X$ is the ambient logarithmic tangent bundle. We prove that five is the smallest positive integer for which this criterion holds for all graphs.