The Lipman-Zariski conjecture in low genus

TL;DR

This paper proves the Lipman-Zariski conjecture for certain low-genus complex surface singularities and characterizes complex 2-tori through properties of their tangent bundles, with broader implications for surface classification.

Contribution

It establishes the conjecture for genus one and two singularities with specific link conditions, and characterizes complex 2-tori via tangent bundle properties, advancing surface singularity theory.

Findings

Proved the conjecture for genus one surface singularities.

Extended results to genus two singularities with non-rational homology sphere links.

Classified smooth projective surfaces with certain tangent bundle properties.

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.