Gordon–Hanamura–Murre's Motivic Decomposition Conjecture

Let $\overline{A}$ and $\overline{X}$ be quasi-projective varieties over $\mathbb C$, with $\overline{A}$ smooth, and let $\overline{f}:\overline{A}\to\overline{X}$ be a projective map. Let

$$\overline{X}=X_0\supset X_1\supset\cdots\supset X_{\dim(X)}$$

be a stratification such that $\overline{f}$ is a stratified map. Write $X_\alpha^0=X_\alpha\setminus X_{\alpha-1}$. Gordon–Hanamura–Murre's Motivic Decomposition Conjecture. There are local systems $\mathcal V^j_\alpha$ on $X_\alpha^0$, a complete set $\Pi^j_\alpha$ of orthogonal projectors, and isomorphisms

$$\sum_{j,\alpha}\Psi^j_\alpha:\mathbb R\overline f_*\mathbb Q_{\overline A}\xrightarrow{\cong}\bigoplus_{j,\alpha}IC_{X_\alpha}(\mathcal V^j_\alpha)[-j-\dim(X_\alpha)]$$

in the derived category. This conjecture seeks a motivic refinement of the decomposition of a projective map into intersection-complex constituents, providing algebraic projectors for the corresponding pieces. The source gives no resolution status, and the general assertion is therefore recorded as open.