Chow motives of genus one fibrations (Q6593267)

This paper (which stems from the author's PhD thesis) is about the Chow motive of surfaces that are ``genus 1 fibrations'', i.e. surfaces \(S\) admitting a fibration \(f\colon S\to C\) over a curve \(C\) such that the generic fibre of \(f\) is a regular genus 1 curve.\N\NThe first main result is as follows:\N\NTheorem: Let \(f\colon S\to C\) be a minimal genus 1 fibration over an algebraically closed field \(k\). Let \(J\to C\) be the associated Jacobian fibration. Then there is an isomorphism\N\[\Nh(X)\cong h(J)\N\]\Nin the category of Chow motives with \({\mathbb{Q}}\)-coefficients. \vskip0.5cm \noindent\N\NThis is a generalization of a result of Coombes about Enriques surfaces [\textit{K. R. Coombes}, Contemp. Math. 126, 47--57 (1992; Zbl 0760.14014)]. This theorem is proven using a decomposition of the Chow motives that was constructed by \textit{J. P. Murre} [J. Reine Angew. Math. 409, 190--204 (1990; Zbl 0698.14032)] and \textit{B. Kahn} et al. [Lond. Math. Soc. Lect. Note Ser. 344, 143--202 (2007; Zbl 1130.14008)], and the argument involves working in the setting of relative motives in the sense of \textit{A. Corti} and \textit{M. Hanamura} [Duke Math. J. 103, No. 3, 459--522 (2000; Zbl 1052.14504)]. A major ingredient in the proof is a result that holds independent interest (Theorem 6.1), concerning the Chow motives of torsors under abelian varieties.\N\NThe second main result reads as follows:\N\NTheorem: Let \(S\) be a smooth projective surface over an algebraically closed field \(k\). Assume that \(S\) has geometric genus zero, and is not of general type. Then \(S\) is Kimura-finite (i.e. \(h(S)\) is finite-dimensional in the sense of [\textit{S.-I. Kimura}, Math. Ann. 331, No. 1, 173--201 (2005; Zbl 1067.14006)]). \N\NThis result generalizes the classical result in characteristic zero of \textit{S. Bloch} et al. [Compos. Math. 33, 135--145 (1976; Zbl 0337.14006)].\N\NThe paper is well-written and largely self-contained: the notions of Chow motives, Chow-Künneth decompositions, Kimura-finiteness, and the classification of surfaces in arbitrary characteristic are helpfully summarized for the reader's benefit.