Remarks on motives of abelian type

Abstract

Let $X$ be a complex smooth projective fourfold with a nef tangent bundle. Then we show that $X$ has a Chow-K"unneth decomposition and that the motivic Lefschetz conjecture holds for $X$. We also show that if $X$ is not the finite quotient of an abelian variety then $X$ satisfies Murre's conjectures. More generally, we establish Murre's conjectures for complex fourfolds whose Chow group of zero-cycles is generated by zero-cycles on a product of curves. Fourfolds with a nef tangent bundle are instances of such fourfolds via a classification result of Demailly-Peternell-Schneider.