Relative Chow-Künneth decompositions for morphisms of threefolds (Q2890363)

Let \(X\) be a smooth projective variety of dimension \(d\) defined over a field \(k\). An absolute Chow-Künneth decomposition for \(X\) consists of a set of mutually orthogonal idempotents \(\pi^{i}_X \in \mathrm{CH}^{d}(X \times X) \otimes \mathbb{Q}\), \(0 \leq i \leq 2d\), adding to the class of the diagonal \(\Delta_X\) and such that the class of \(\pi^i_X\) in \(H^{2d}(X\times X,\mathbb{Q})\) is the Künneth projector on \(H^i(X, \mathbb{Q})\). This notion of Chow-Künneth decomposition is relevant because it fits into the conjectural framework of Bloch and Beilinson for algebraic cycles. Curves and, as shown by \textit{J. P. Murre} [J. Reine Angew. Math. 409, 190--204 (1990; Zbl 0698.14032)], surfaces have a CK decomposition. There is no general result in higher dimensions although certain varieties are known to have a CK decomposition, e.g. certain threefolds, varieties whose motive is of abelian type or varieties with small Chow groups.NEWLINENEWLINELet \(f: X \rightarrow S\) be a surjective projective morphism of complex algebraic varieties. A relative CK decomposition for \(f\) consists of mutually orthogonal idempotents \(\pi^{i}_{X/S} \in \mathrm{CH}_{\dim X}(X \times_S X) \otimes \mathbb{Q}\) adding to the relative diagonal \(\Delta_{X/S}\) and such that their action on the perverse cohomology sheaf is a suitable projector. The main result of this paper describes criteria on a non-constant surjective projective morphism \(f: X \rightarrow S\) of quasi-projective complex varieties with \(X\) \(3\)-dimensional and smooth for \(f\) to admit a relative CK decomposition.NEWLINENEWLINEAs a corollary, the authors establish sufficient conditions on a smooth projective complex \(3\)-fold \(X\) to have an absolute CK decomposition. On the one hand, this gives another proof of a previous result of the first author and del Angel, and, on the other hand, this improves a previous result of the second author.