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THE LIPMAN–ZARISKI CONJECTURE IN GENUS ONE HIGHER

Part of: Singularities Local theory Differential algebra Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  23 April 2020

HANNAH BERGNER  and
PATRICK GRAF
HANNAH BERGNER
Affiliation:
Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Straße 1, 79104Freiburg im Breisgau, Germany; Hannah.Bergner-c9q@ruhr-uni-bochum.de
PATRICK GRAF
Affiliation:
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT84112, USA; patrick.graf@uni-bayreuth.de

Abstract

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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.

MSC classification

Primary: 14B05: Singularities
Secondary: 14J17: Singularities 32S25: Surface and hypersurface singularities 13N15: Derivations
Type
Algebraic and Complex Geometry
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e21
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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