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Differential forms on log canonical spaces. (English) Zbl 1258.14021
This monumental paper is concerned with differential forms on singular varieties. The classes of singularities treated are those that naturally arise in the Minimal Model Program, that is Kawamata log terminal singularities (klt) and divisorial log terminal singularities (dlt). The main question is whether a \(p\)-form on the smooth locus of a variety \(X\) of dimension \(n\) extends regularly to any resolution of singularities for \(p\leq n\). This can be rephrased as follows. Let \(f:Y\to X\) be a resolution of singularities then the question is whether the push forward sheaf \(f_*\Omega^p_Y\) is reflexive for \(p\leq n\). The authors are able to prove that this is the case for klt pairs via a more general result established for log canonical pairs. With the machinery involved in the proof and much more they also provide a theory of differential forms on dlt varieties. Aside this huge work applications are given to vanishing theorems and Lipman–Zaroski conjecture on the ssmoothness of varieties with locally free tangent sheaf.

MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C38 Sheaves of differential operators and their modules, \(D\)-modules
14B05 Singularities in algebraic geometry
32S20 Global theory of complex singularities; cohomological properties
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