Publications mathématiques de l'IHÉS

, Volume 114, Issue 1, pp 87-169

First online:

Differential forms on log canonical spaces

  • Daniel GrebAffiliated withMathematisches Institut, Albert-Ludwigs-Universität Freiburg Email author 
  • , Stefan KebekusAffiliated withMathematisches Institut, Albert-Ludwigs-Universität Freiburg
  • , Sándor J KovácsAffiliated withDepartment of Mathematics, University of Washington
  • , Thomas PeternellAffiliated withMatematisches Institut, Universität Bayreuth

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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.