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A complex Mixed Hodge Structure is given by a complex vector space $V$ together with a descending filtration $W$ and two ascending filtrations $F,\bar{F}$ that satisfy the condition

\begin{equation} gr_F^pgr_{\bar{F}}^qgr_n^W(V)=0\qquad\text{if }n\neq p+q \end{equation}

It is a result of Deligne that this is the equivalent to the data of a bigraded vector space $\tilde{V}$, together with an automorphism $\delta$ satisfying

\begin{equation} (\delta-1)(\tilde{V}^{p,q})\subset\bigoplus_{p'<p,q'<q}\tilde{V}^{p',q'} \end{equation}

Is there a version of this that carries over to families, i.e. can a variation of Mixed Hodge Structure also be described by something like a (maybe $C^{\infty}$-) bundle with a decomposition into subbundles and a bundle automorphism? If yes, how does Griffiths transversality enter the picture?

Remark: I posted this on MSE before, where it was not answered.

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I guess you're refering to the main result of Deligne "Structure de Hodge mixtes réelles". Given a real (resp. complex) variation of mixed Hodge structures consisting of a holomorphic bundle $V$, integrable connection $\nabla$, flat real sub bundle $W_\bullet$, holomorphic sub bundles $F^\bullet$ (and antiholomorphic bundles $\bar F$), you would have a $C^\infty$ bigrading $V=\bigoplus V^{pq}$ with $$V^{pq} = F^pGr^{p+q}_WV\cap \bar F^q Gr^{p+q}_WV$$ Although I haven't checked carefully, it seems that his formula gives a $C^\infty$ bundle automorphism $\delta$ satisfying the condition you stated. So perhaps this is positive answer. On the other hand, this wouldn't be an equivalence, in general. I don't think you can recover the holomorphic structure on $V$, or connection from this information.

Further comments To get some sense of the problem, start with the case of a pure variation. So now $\delta=0$, and all you have is a bigrading, which is not much information at all. However, if you also specify the monodromy, than you can basically recover everything else, by work of Simpson over a projective base, or T. Mochizuki in general. For some extensions to the mixed case, you can take a look at some papers of Pearlstein, e.g. Degenerations of mixed Hodge structure, Duke (2001).

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  • $\begingroup$ thank you for the answer. Yes it is exactly this paper I was referring to and your observations are basically what motivated the question (which was maybe too vaguely posed). I wonder if this can be made an equivalence by adding further information and which this should be. If you don't mind I will leave the question open for some time to see if there are further ideas or if something is known about this and otherwise accept you answer in a few days. $\endgroup$
    – jorst
    Dec 3, 2015 at 14:31
  • $\begingroup$ Thanks for the additional comments (Due to the mechanics of the site I was not noticed and just saw them now). I'll look at the paper of Pearlstein. Could you be so kind to include references to the mentioned works of Simpson and Mochizuki? $\endgroup$
    – jorst
    Dec 11, 2015 at 13:36

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