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 is a guess as to what the "suitable sense" should be, inspired by the case when we have the weight filtration alone. Given a morphism $X\to Y$ of two varieties and given a minimal model $M_X\to A_{PL}(X), M_Y\to A_{PL}(Y)$ for each variety, there is a map of the minimal models, unique up to a homotopy preserving both filtrations and such that the obvious diagram commutes up to homotopy. (This would imply the uniqueness of the mHs on the minimal model.)
Here is a related (and maybe equivalent?) question: due to Kadeishvili, the cohomology $H^\ast$ of any cdga carries an $A_{\infty}$ structure, and in fact, a $C_{\infty}$ structure. See e.g. Keller, Introduction to $A_\infty$ algebras and modules, theorem on p. 7. If now $H^\ast$ is the cohomology of a complex algebraic variety, can one choose the structure maps $(H^\ast)^{\otimes n}\to H^*$ to be morphisms of mixed Hodge structures? Once again, this should be functorial in an appropriate sense.
Here are some remarks:
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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.
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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. -
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.
Best Answer
I think these questions are very satisfactorially answered in a paper of joana cirici on arxiv 2013. this paper was published in 2015 dennis sullivan