Differential forms in algebraic geometry -- a new perspective in the singular case
type: Item
instance of: scholarly article
title: Differential forms in algebraic geometry -- a new perspective in the singular case
zbMATH Open document ID: 1359.14001
DOI: 10.4171/PM/1990
published in: Portugaliae Mathematica. Nova Série
publication date: 2016-12-30T00:00:00Z
review text: 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.
reviewed by: Federico Buonerba
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 6669856
zbMATH Keywords: algebraic differential forms
zbMATH Keywords: cohomological invariants
zbMATH Keywords: h-topology
zbMATH Keywords: singular varieties
Wikidata QID: Q115211941
author: Annette Huber-Klawitter
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/pm/1990
OpenAlex ID: W2557993511
description: scientific article; zbMATH DE number 6669856