On the v-Picard Group of Stein Spaces

Abstract We study the image of the Hodge–Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of close...

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Bibliographic Details
Published in:International mathematics research notices 2024-10, Vol.2024 (20), p.13352-13379
Main Authors: Ertl, Veronika, Gilles, Sally, Nizioł, Wiesława
Format: Article
Language:English
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Summary:Abstract We study the image of the Hodge–Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of closed differential forms. We show that it indeed always contains such forms but the quotient can be non-trivial: it contains a slightly mysterious $\mathbf{Z}_{p}$-module that maps, via the Bloch–Kato exponential map, to integral classes in the pro-étale cohomology. This quotient is already non-trivial for open unit disks of dimension strictly greater than $1$.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnae185