Motivic decomposition and intersection Chow groups. II.
Let \(X\), \(S\) be quasi-projective complex algebraic varieties, with \(X\) smooth and let \(p: X\to S\) be a projective map. The direct image of the constant sheaf \({\mathcal R}p_*\mathbb{Q}_X\) is a direct sum of intersection complexes. Conjecturally it should be possible to lift such a decomposition to a suitably defined motivic category \({\mathcal M}\). In the case \({\mathcal M}\) is the category \({\mathcal C}H(S)\) of Chow motives over \(S\), then the existence of such a decomposition is a consequence of the conjectures of Grothendieck, Bloch-Beilinson-Murre, as it has been proved by the same authors in [Duke Math. J. 103, 459--522 (2000; Zbl 1052.14504)]. In this paper the authors consider the case when the map \(p: X\to S\) is a resolution of singularities. Then one of the direct summands of \({\mathcal R}p_*\mathbb{Q}_X\) is the intersection complex \(IC_S\) of \(S\); the corresponding direct summand in the motivic decomposition is the motivic intersection complex of \(S\) and its Chow group is denoted by \(ICH^r(S)\). Assuming the same conjectures as above, so that a motivic decomposition does exist, the group \(ICH^r(S)\) is a canonical subquotient of the Chow group \(CH^r(X)\). Moreover \(CH^r(X)\) has a decreasing filtration \(F^\bullet_S\) such that, if \(f: X\to Y\) is a projective surjective map then the injection \(f*: CH^r(Y)\to CH^r(X)\) and the surjection \(f_*: CH_s(X)\to CH_s(Y)\) are strictly compatible with the filtrations. On the other hand one can define a filtration \(F^\bullet_S\) on \(CH_s(X)\) satisfying suitable properties, similar to the ones introduced in [loc.cit.] for the case \(S=\text{Spec } k\). Then, without assuming the existence of a motivic decomposition, hence without assuming the conjectures, the authors give an ``unconditional'' definition of intersection Chow groups, independent of the choice of a resolution.
scientific article; zbMATH DE number 5296031