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?

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