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Differential forms are a rich source of invariants in algebraic geometry. This approach was very successful for smooth varieties, but the singular case is less well-understood. We explain how the use of the h-topology (introduced by Suslin and Voevodsky in order to study motives) gives a very good object also in the singular case, at least in characteristic 0. We also explain problems and solutions in positive characteristic.
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Annette Huber-Klawitter, Differential forms in algebraic geometry – A new perspective in the singular case. Port. Math. 73 (2016), no. 4 pp. 337–367