We prove the Lipman–Zariski conjecture for complex surface singularities with$p_{g}-g-b\leqslant 2$. Here$p_{g}$is the geometric genus,$g$is the sum of the genera of 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 a 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.

中文翻译：

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LIPMAN-ZARISKI 猜想高一属

我们证明了复杂表面奇点的 Lipman-Zariski 猜想$p_{g}-gb\leqslant 2$. 这里$p_{g}$是几何属，$g$是异常曲线的总和，并且$b$是对偶图的第一个 Betti 数。这比第二作者的先前结果有所改进。作为一个应用，我们展示了一个具有局部自由切层的紧致复曲面是光滑的，只要它承认两个一般线性独立的扭曲向量场，并且它的规范层最多具有两个全局截面。