# in which sense is a mixed Hodge structure an extension of pure ones?

I have heard several times that mixed Hodge structures are iterated extensions of pure ones. What does it mean? Here is what I figured out.

A mixed Hodge structure $H$ comes with an increasing filtration

$W_{-1} H =(0) \subset W_0H \subset W_1H \subset \cdots \subset W_mH=H$

For each $k$, the inclusion $W_{k-1} \subset W_k$ gives an exact sequence $$0 \to W_{k-1}H \to W_kH \to Gr_k^W H:=W_k H/W_{k-1} H \to 0$$ which is, I guess, compatible with the Hodge structures, so $W_kH$ is an extension of $Gr_k^WH$ (which is pure) by $W_{k-1}H$ (which is not).

In particular, $H$ is an extension of $Gr_m^W H$ by $W_{m-1}H$, $W_{m-1}H$ is an extension of $Gr_{m-1}^W H$ by $W_{m-2}H$ and so on until $W_0H$ which is a pure Hodge structure.

Is that right? Am I missing something?

• It is necessary that $W_{-1}=0$ but everything else is correct. Jun 15, 2014 at 19:21
• I meant "it isn't necessary..." Jun 15, 2014 at 19:35
• Oh yeah, I was thinking in the case $H$ is the cohomology of an algebraic variety, so weights are positive. But of course the same could be apply to $W_{-n}H=(0) \subset ....$ for $n$ large enough Jun 15, 2014 at 19:55
• You are not missing anything. The term iterated extension means that (in your example): $W_1$ is an extension of $Gr_1$ by $W_0$, $W_2$ is an extension of $Gr_2$ by $W_1$ etc. So $H$ is "built up" from pure modules, in a similar way that a finite group is built as an iterated extension of finite simple groups. Jun 15, 2014 at 23:01

You are not missing anything. The term iterated extension means that (in your example): $W_{1}$ is an extension of $\mathrm{Gr}_{1}$ by $W_{0}$, $W_{2}$ is an extension of $\mathrm{Gr}_{2}$ by $W_{1}$ etc. So $H$ is "built up" from pure modules, in a similar way that a finite group is built as an iterated extension of finite simple groups.
It isn't necessary that $W_{−1}=0$ but everything else is correct.