# Mixed Hodge structure on the rational homotopy type

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:

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.

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Have you looked at the papers of Hain: The de Rham homotopy theory of algebraic varieties I, II? They are certainly related to your question and even if they do not contain an answer they might still be useful. – ulrich Apr 23 '10 at 14:35
unknown -- thanks, yes, I've seen these and they are useful, and the statement I'm interested in is mentioned in the first paper (theorem 3 in the first paper). But the proof is by "plugging (6.2.1) into Morgan's machine". It is not stated explicitly where we are supposed to plug (6.2.1), but I have a guess and if my guess is correct, then this wouldn't work, due to example in remark 3 above. – algori Apr 23 '10 at 20:00
In Hain's original research announcement, Mixed Hodge structures on homotopy groups, Bull. LMS 14 (1986) 11-114, Theorem 3 states that the minimal model of a complex variety carries a MHS, but it is not necessarily unique. I suppose you would like to add some kind of uniqueness to this statement. – Jeffrey Giansiracusa May 5 '10 at 15:53
Jeffrey -- thanks, did you mean Bull. AMS? As to the uniqueness: I have a stronger (but a bit vague) requirement "functorial in a suitable sense". Depending on that the "suitable sense" is, this may (or may not) imply the uniqueness. Probably I should make this more explicit in the posting. – algori May 9 '10 at 22:19
oops, yes, I meant Bull AMS. – Jeffrey Giansiracusa May 10 '10 at 9:02