The Lipman–zariski Conjecture in Genus One Higher

Abstract

We prove the Lipman-Zariski conjecture for complex surface singularities of genus at most two, and also for those of genus three whose link is not a rational homology sphere. This improves on a previous result of the second author. As an application, we show that a compact complex surface with locally free tangent sheaf is smooth as soon as it admits two generically linearly independent twisted vector fields and its canonical sheaf has at most two global sections.