# The Lipman-Zariski conjecture in genus one higher

## TL;DR

This paper proves the Lipman-Zariski conjecture for certain complex surface singularities, showing that under specific conditions, such singularities are smooth, and applies this to characterize smoothness of complex surfaces with particular vector fields.

## Contribution

It extends the Lipman-Zariski conjecture's validity to a broader class of surface singularities with specific invariants, improving previous results.

## Key findings

- Proved the conjecture for singularities with $p_g - g - b \,\leq 2$.
- Showed that certain complex surfaces with specific vector fields are smooth.
- Established conditions under which surface singularities are smooth.

## Abstract

We prove the Lipman-Zariski conjecture for complex surface singularities with $p_g - g - b \le 2$. Here $p_g$ is the geometric genus, $g$ is the sum of the genera of the exceptional curves and $b$ is the first Betti number of the dual graph. 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.