Chow motives of abelian type over a base
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)].
scientific article; zbMATH DE number 7073899
scientific article