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Differential forms on log canonical spaces

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Abstract

The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting.

Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces.

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Authors and Affiliations

  1. Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104, Freiburg im Breisgau, Germany

    Daniel Greb & Stefan Kebekus

  2. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195, USA

    Sándor J Kovács

  3. Matematisches Institut, Universität Bayreuth, 95440, Bayreuth, Germany

    Thomas Peternell

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  1. Daniel Greb
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  2. Stefan Kebekus
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  3. Sándor J Kovács
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  4. Thomas Peternell
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Correspondence to Daniel Greb.

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In memory of Eckart Viehweg

Daniel Greb was supported in part by an MSRI postdoctoral fellowship during the 2009 special semester in Algebraic Geometry. Stefan Kebekus and Thomas Peternell were supported in part by the DFG-Forschergruppe “Classification of Algebraic Surfaces and Compact Complex Manifolds”. Sándor Kovács was supported in part by NSF Grants DMS-0554697 and DMS-0856185, and the Craig McKibben and Sarah Merner Endowed Professorship in Mathematics.

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Greb, D., Kebekus, S., Kovács, S.J. et al. Differential forms on log canonical spaces. Publ.math.IHES 114, 87–169 (2011). https://doi.org/10.1007/s10240-011-0036-0

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