  We show that geometric syntomic cohomology lifts canonically to the category
of Banach-Colmez spaces and study its relation to extensions of modifications
of vector bundles on the Fargues-Fontaine curve. We include some computations
of geometric syntomic cohomology Spaces: they are finite rank $Q_p$-vector
spaces for ordinary varieties, but in the nonordinary case, these cohomology
Spaces carry much more information, in particular they can have a non-trivial
$C$-rank. This dichotomy is reminiscent of the Hodge-Tate period map for
$p$-divisible groups.
