Differential forms in algebraic geometry -- a new perspective in the singular case
In this survey article, the problem of defining differential forms on singular varieties is considered. The first four chapters serve as an introduction, while in chapter 5 the approach of the authors and her collaborators is outlined: the idea is to consider differentials as a sheaf in a Grothendieck topology, where every variety is locally non-singular -- such as Voevodsky's h-topology, or sub-topologies thereof avoiding inseparable covers that would lead to pathologies in positive characteristic. Requiring that differentials behave as usual on non-singular varieties, this trick provides a natural and functorial extension of this notion, to the realm of singular varieties. Chapter 6 is devoted to a summary of the properties of differentials, in characteristic zero. It turns out that a differential on a singular variety is completely determined by a collection of compatible differentials on non-singular h-covers. Chapter 7 is focused on positive characteristic phenomena, where of course a major issue is the lack of a resolution of singularities. The suggested solution, inspired by the characteristic zero situation, is to work with Riemann-Zariski spaces of valuations, and to define a differential on a singular variety as a compatible collection of differentials defined on spectra of fields and valuation rings mapping to our variety. Finally, some conjectures are proposed.
scientific article; zbMATH DE number 6669856