Is there a description of variation of (mixed) Hodge Structures in terms of a Deligne operator?

Question

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.

Answer 1

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).