I have two (not unrelated) questions. Let me first give a short introduction.


Introduction
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For a general overview of the setup I refer to the introduction (§1) of [Zhang].


Let $k$ be a number field or a function field of a curve. Let $X/k$ be a curve, and assume that $X$ has a semistable model. Let $Y$ be the triple product $X^{3}$.


In [§5.1 (pp. 77v.), Zhang] he defines a motive $M$ as the kernel of the map
$$ \bigwedge^{3} H^{1}(X)(2) \to H^{1}(X)(1) \colon a \wedge b \wedge c \mapsto
a(b \smile c) + b(a \smile c) + c(a \smile b). $$
I think I have figured out what he means, and why this is a Chow motive.


A little further, he defines $\text{Ch}(M)$ as a subgroup of the kernel of the cycle map $\text{Ch}^{2}(Y) \to H^{4}(Y)$. It is the subgroup of elements $z$ satisfying


* $z$ is stable under the action of the symmetric group $S\_{3}$ on $Y$;
* the push-forward $\text{pr}\_{12,\*} z$ equals $0$, where $\text{pr}\_{12}$ is the projection on the first two coordinates;
* if $i$ denotes the embedding $X^{2} \to Y \colon (x,y) \mapsto (x,x,y)$, and $\text{pr}\_{2} \colon X^{2} \to X$ the projection on the second coordinate, then $\text{pr}\_{2,\*}i^{\*} z = 0$.


Questions
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**Q1.** Why *this* definition?


**Q2.a.** The notation suggests to me that there is a more general definition of $\text{Ch}(M)$, for Chow motives $M$. Is this true?


**Q2.b.** If so, can someone give me pointers (reference request) to a definition of $\text{Ch}(M)$, since all my Googles for "Chow group motive" give me results that use Chow groups to define motives (and I am actually not surprised to get those results.


I think that if $M$ is the motive corresponding to a smooth projective variety $X$, then $\text{Ch}(M)$ should equal $\text{Ch}(X)$. But what about the other motives?




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References
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[Zhang] Shou-Wu Zhang. *“Gross–Schoen Cycles and Dualising Sheaves”*. url: <http://arxiv.org/0812.0371.pdf>.



Chow group of a (particular) motive [+ reference request]
One can define the Chow groups of a Chow motive $M$ by the formulas
$$ \newcommand{\Hom}{\mathrm{Hom}}\newcommand{\L}{\mathbb{L}} \newcommand{\Ch}{\mathrm{CH}} \Ch^s(M) = \Hom(M,\L^s) $$
and 
$$ \Ch\_s(M) = \Hom(\L^s,M).$$
Here $\L = H^2(\mathbf P^1).$ A reference for this definition is e.g. Scholl's text in the first Motives volume. When $M$ is the motive of a smooth projective variety $X$, then $\Ch^s X = \Ch^s M$ etc. But I must confess that I don't see why the definition of $\mathrm{Ch}(M)$ given in the question is consistent with this. 


