For smooth proper schemes over $\mathbb{C}_p$, there is no canonical Hodge--Tate decomposition (but there is something close). Is there an analogue of this on the archimedean side? I thought about this a little bit but did not arrive anywhere.

## 1 Answer

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See Example 4.1.2 in the notes of Brinon--Conrad. There is something similar going on there: for a smooth proper scheme over complex numbers there is a functorial splitting of the Hodge filtration, but for an arbitrary field of characteristic 0 there is only an exhaustive and separated filtration whose associated graded vector space is the Hodge cohomology (and it does not admit a functorial splitting, generally speaking).