Let $P$ be a smooth projective variety. For an object $X$ of $D_{perf}(P)$ (i.e. a bounded perfect complex of $\mathfrak{O}_P$-module sheaves) we consider its class $[X]$ in $K_0(P)\otimes \mathbb{Q}(\cong K_0 (D_{perf}(P))\otimes \mathbb{Q}) $. Now, this $K_0$-group is isomorphic to $\bigoplus_{0\le i\le \dim P} Chow_i(P)$. My question is: are there any 'nice' restrictions on $X$ that ensure that $[X]$ is 'pure' i.e. that it belongs to $ Chow_j(P)\subset\bigoplus_{0\le i\le \dim P} Chow_i(P)$ (for some fixed $j$). Actually, I am interested in the case $P=A\times B$, $j=\dim A$.
Unfortunately, my question is somewhat vague. I do not want the condition in question to be too restrictive; also I don't want it to be stated in terms of $[X]$.
P.S. Instead, it is certainly sufficient to understand the conditions $X\in \bigoplus_{0\le i\le j} Chow_i(P)$ and $[X]\in \bigoplus_{j\le i\le \dim P} Chow_i(P)$. The latter one seems to be especially mysterious; is there any procedure (duality?) that reduces it to the first one?
