Does there exist a functorial splitting for the weight filtration (of singular cohomology)? There are plenty of examples of varieties whose singular cohomology with rational coefficients considered as a mixed Hodge structure does not decompose as the direct sum of its pure (weight) factors. Yet if we consider the cohomology just as filtered vector spaces over rationals, such a decomposition certainly exist (for any variety). 
My question is: could there exist a functorial decomposition like this (say, for the singular cohomology as a functor from the category of all smooth complex varieties), or does there exist some obstruction for such a functorial splitting?
 A: Mikhail,
This is an interesting question. But I think that  the answer is no, 
there would be no functorial splitting
of the weight filtration as filtered vector spaces. 
This needs a bit of work perhaps, but here is my example. Let $E$ be an elliptic curve.
Let the $\sigma$ the involution given by $-1$ in the group law. Choose a non $2$-torsion point
$p$. Then $q=\sigma(p)\not= p$. By duality, we may work with homology. 
Choose small loop $\gamma_p$ about $p$ and let $\gamma_q$ be its image under $\sigma$.
Note that the class $[\gamma_p+\gamma_q]=0$ in $H_1(E-\{p,q\})$.
Then there is an exact
sequence
$$\mathbb{Q}^2\to H_1(E-\{p,q\})\to H_1(E)\to 0$$
where the first map sends  $(a,b)$ to $a[\gamma_p]+b[\gamma_q]$. 
A splitting  would send $H_1(E-\{p,q\})$ to 
$W_0=span(\gamma_p)$, or to  the anti-invariant
part of $\mathbb{Q}^2$ under $\sigma$. However, functoriallity  should  imply 
that the splitting ought to be invariant.
Added This example is a bit fishy as it stands (see comments) but I think the
basic strategy should work. I'll try to fix it in the morning. 
A: There is the Deligne splitting. I take this from Peters and Steenbrink's book, Section 3.1.
For a complex variety $X$, we define $I^{p,q} \subseteq H^{\ast}(X, \mathbb{C})$ by
$$I^{p,q} := F^p \cap W_{p+q} \cap \left( \overline{F}^q \cap W_{p+q} + \sum_{j \geq 2} \overline{F^{q-j+1}} \cap W_{p+q-j} \right).$$
Then $H^{\ast}(X, \mathbb{C}) = \bigoplus I^{p,q}$ and $W_k \otimes \mathbb{C} = \bigoplus_{p+q \leq k} I^{p,q}$ and $F^p = \bigoplus_{r \geq p} \bigoplus_s I^{r,s}$.
In particular, defining $U_k = \bigoplus_{p+q=k} I^{p,q}$ gives a splitting of the weight filtration tensored with $\mathbb{C}$. If I am not mistaken, it is functorial.
The Deligne splitting only exists with $\mathbb{C}$ coefficients. I think Donu Arapura's answer here is fairly convincing that there is no splitting with $\mathbb{Q}$ or $\mathbb{R}$ coefficients. 
Disclaimer: I just learned about this today, so I might be missing something.
