On the Chow motive of some moduli spaces (Q2706828)

The author studies the Chow motive of moduli spaces of stable vector bundles over a smooth projective curve. In particular, he proves that this motive lies in the category generated by the curve, and he computes its class in the Grothendieck ring of the category of motives. These results rely on geometric constructions due to \textit{E. Bifet, F. Ghione} and \textit{M. Letizia} [Math. Ann.~299, 641--672 (1994; Zbl 0840.14003)], and on the author's computation of the Chow motive of a smooth projective variety acted on by the multiplicative group \({\mathbb{G}}_m\) which in turn relies on the work of \textit{A. Białynicki-Birula [Ann. Math.~(2)~98, 480--497 (1973; Zbl 0275.14007)], and which may be considered as a generalization of the reviewer's earlier result \textit{B. Köck} [Manuscr. Math.~70, 363--372 (1991; Zbl 0735.14001)].NEWLINENEWLINEAs applications the author computes the Poincaré-Hodge polynomial and the number of points over a finite field of these moduli spaces. In particular, he recovers a result of \textit{M. S. Narasimhan} and \textit{S. Ramanan} on the \(\chi_y\)-genus [J. Indian Math. Soc., New Ser.~39, 1--19 (1975; Zbl 0422.14018)].NEWLINENEWLINEFinally the author studies some well-known conjectures on algebraic cycles on these moduli spaces, namely the Hodge, Tate and standard conjectures. For instance, he proves that the standard conjecture of Lefschetz type B holds for these moduli spaces, a result which, over the field of complex numbers, has previously been proved by \textit{I. Biswas} and \textit{M. S. Narasimhan} [J. Algebr. Geom.~6, 697--715 (1997; Zbl 0891.14002)].}