Geodesic
THE LIPMAN–ZARISKI CONJECTURE IN GENUS ONE HIGHER
Forum of Mathematics, Sigma, Tome 8 (2020)

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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. 
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     author = {HANNAH BERGNER and PATRICK GRAF},
     title = {THE {LIPMAN{\textendash}ZARISKI} {CONJECTURE} {IN} {GENUS} {ONE} {HIGHER}},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
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     doi = {10.1017/fms.2020.19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2020.19/}
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HANNAH BERGNER; PATRICK GRAF. THE LIPMAN–ZARISKI CONJECTURE IN GENUS ONE HIGHER. Forum of Mathematics, Sigma, Tome 8 (2020). doi: 10.1017/fms.2020.19

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