The Lipman-Zariski conjecture in low genus

Abstract

We prove the Lipman-Zariski conjecture for complex surface singularities of genus one, and also for those of genus two whose link is not a rational homology sphere. As an application, we characterize complex $2$-tori as the only normal compact complex surfaces whose smooth locus has trivial tangent bundle. We also deduce that all complex-projective surfaces with locally free and generically nef tangent sheaf are smooth, and we classify them.

Published version is slightly shortened

References in corpus (1)

• The Zariski-Lipman conjecture for log canonical spaces

Cited by in corpus (1)

• The Lipman-Zariski conjecture in genus one higher