A mixed Hodge structure (mHs) on a commutative differential graded algebra (cgda) over $\mathbf{Q}$ is a mixed Hodge structure on the underlying vector space such that the product and the differential are maps of mixed Hodge structures. If a cdga is equipped with an mHs, then so is its cohomology. Now let $A$ be, say, the algebra of piecewise polynomial cochains of a complex algebraic variety $X$ and let $M$ be the minimal model of $A$. Is there a mixed Hodge structure on $M$ which would be functorial in some sense and such that the comparison quasi-isomorphism $M\to A$ induces a map of mixed Hodge structures?

Here are some remarks:

1. If one considers weight filtrations alone, then two things happen. First, one has to slightly modify the above definition: a weight filtration on a cdga is a filtration such that the differential and the product are strictly compatible with it (this is automatic in the mixed Hodge case). Second, the answer to the similar question is yes: namely, there is a weight filtration on the minimal model of an algebraic variety, it induces the ``right'' weight filtration in cohomology and the functoriality property can be stated as follows: a map of algebraic varieties gives a map of filtered minimal models, unique up to filtration-preserving homotopy. This is essentially due to J. Morgan, The algebraic topology of smooth algebraic varieties, Publ. IHES 48, 1978 and Navarro-Aznar, Sur la th'eorie de Hodge-Deligne, Inv. Math. 90, 1987.

2. In the above-mentioned paper Navarro-Aznar claims (7.7, p.38) that the answer to the question of this posting is positive as well. He does not give a proof however, instead referring the reader to Morgan (ibid) and ``la deuxi`eme partie de cet article''. It is not clear to me how to deduce the statement (if it is true) from either one of these sources.

3. Let $X$ be a smooth complete curve and let $K$ be a finite subset with $\geq 2$ elements. Then according to Morgan, a filtered model of $X-K$ is the $E_2$-term of the Leray spectral sequence of the embedding $X-K\subset X$, equipped with the Leray filtration. This indeed determines the cohomology of $X-K$ as a filtered algebra. But the $E_2$-term viewed as a mixed Hodge structure does not determine $H^\ast(X-K,\mathbf{Q})$ as a mixed Hodge structure since it does not depend on $K$ and $H^\ast(X-K,\mathbf{Q})$ does.