Differential Forms on Log Canonical Spaces

Details Differential Forms on Log Canonical Spaces Differential Forms on Log Canonical Spaces

2010-03-15

arxiv.org/abs/1003.2913v4

Publ. Math. IHES 114 (2011), no. 1, 87-169

Mathematics

Algebraic Geometry

72 pages, 6 figures. A shortened version of this paper has appeared in Publications math\'ematiques de l'IH\'ES. The final pub

Scientific paper

10.1007/s10240-011-0036-0

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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