The conjecture 51 in Levine's Mixed Motives in Handbook of K-theory


<http://www.math.illinois.edu/K-theory/handbook/1-429-522.pdf>


states that the functor induced by $hs:\text{ECM} \rightarrow \text{MHS}$, denoted by,
\begin{equation}
\mathfrak{hs}:\text{NMM}(k)\_{\mathbb{Q}} \rightarrow \text{MHS}\_{\mathbb{Q}}
\end{equation}
where $\text{NMM}(k)\_{\mathbb{Q}}$ is Nori's mixed motives over a field $k$ (which admits an embedding $\sigma: k \rightarrow \mathbb{C}$) tensored by $\mathbb{Q}$. I am not clear with the construction of the functor $\mathfrak{hs}$ (I guess this is the Hodge realisation functor of Nori's mixed motives), could anyone give me some references? I also have two questions about the properties of $\mathfrak{hs}$.


1. Is $\mathfrak{hs}$ (conjectured to be) exact or not? Has the exactness of it been proved?
2. If $\mathfrak{hs}$ is exact, then it has a derived functor,
\begin{equation}
D\mathfrak{hs}:D^b(\text{NMM}(k)\_{\mathbb{Q}}) \rightarrow D^b(\text{MHS}\_{\mathbb{Q}})
\end{equation}
In Harrer's thesis,


<https://arxiv.org/abs/1609.05516>


he constructs a functor,


\begin{equation}
R\_n:\text{DM}\_{gm}(k,\mathbb{Q}) \rightarrow D^b(\text{NMM}(k)\_{\mathbb{Q}})
\end{equation}
The composition of it with $D\mathfrak{hs}$ is a functor 


\begin{equation}
D\mathfrak{hs} \circ R\_n :\text{DM}\_{gm}(k,\mathbb{Q}) \rightarrow D^b(\text{MHS}\_{\mathbb{Q}})
\end{equation}
Is this functor equivalent to the Hodge realisation functor? Any references?



Nori's Mixed Motives and Realisation Functors
Did you read <http://home.mathematik.uni-freiburg.de/arithgeom/preprints/buch/buch-v1.pdf>?


About 1: since the forgetful functor $MHS\_{\mathbb{Q}}\to \mathbb{Q}-Vect$ is exact and faithful, it suffices to verify that the singular cohomology functor from Nori motives is exact. The latter statement should be an immediate consequence of the definition of Nori motives. 


