I will report on a joint project with S. Gilles and W. Nizioł studying the image of the Hodge-Tate logarithm map (defined by Heuer) in the case of smooth Stein varieties.
Motivated by the computations for the affine space, Heuer raised the question whether this image is always equal to the group of closed differential forms.
We show that it always contains such forms but the quotient can be non-trivial.
More precisely, it contains a Z_p-module which maps to integral classes in the proétale cohomology via the Bloch-Kato exponential map.