Chow motives of abelian type over a base
type: Item
instance of: scholarly article
title: Chow motives of abelian type over a base
zbMATH Open document ID: 1422.14012
published in: European Journal of Mathematics
publication date: 2019-06-27T00:00:00Z
review text: Given a reasonable base \(S\) one can define the category of Chow motives over \(S\). The paper under review studies the motive of a family of curves \(C\) over \(S\) and the motive of its relative Jacobian \(J\). The main result shows that the motives of \(C\) and \(J\) are both Kimura finite dimensional. The argument goes through a geometric description of \(J\) in term of the symmetric products of \(C\) (following the classical case where \(S\) is the spectrum of a field). From the basic properties of Kimura finiteness one deduces that all objects in the pseudo-abelian tensor category generated by motives of curves (or equivalently Jacobians) are finite dimensional. The author decide to call these objects motives of abelian type. This name is misleading as motives of Jacobians are in general not enough to generate all motives of abelian schemes. (Over a field it is the case as any simple abelian variety appears as a quotient of a Jacobian. This fact does not hold in the relative setting.) It turns out that the result for all abelian schemes was already known, see for example [\textit{P. O'Sullivan}, J. Reine Angew. Math. 654, 1--81 (2011; Zbl 1258.14006)]. More recently it has been shown that motives of any smooth commutative group scheme are finite dimensional [\textit{G. Ancona} et al., Algebr. Geom. 3, No. 2, 150--178 (2016; Zbl 1354.14011)].
reviewed by: Giuseppe Ancona
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14G40
zbMATH DE Number: 7073899
zbMATH Keywords: relative curves
zbMATH Keywords: Jacobian scheme
zbMATH Keywords: Picard bundles
zbMATH Keywords: symmetric powers
zbMATH Keywords: Riemann-Roch theorem
zbMATH Keywords: Chow motives
zbMATH Keywords: Chern class
zbMATH Keywords: projective bundle theorem
zbMATH Keywords: traces
zbMATH Keywords: numerical equivalence
zbMATH Keywords: nilpotent correspondences
author: V. Guletskiĭ
MaRDI profile type: Publication
full work available at URL: s40879-018-0270-9
OpenAlex ID: W2888004403
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cites work: Symmetric Products and Jacobians
cites work: Picard bundles
cites work: Q2551490
cites work: Jacobians and symmetric products
cites work: Q2546548
cites work: Q5795562
DOI: 10.1007/S40879-018-0270-9
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description: scientific article; zbMATH DE number 7073899