There is no Definable Grauert Direct Image Theorem
Algebraic Geometry
2026-02-16 v2 Logic
Abstract
We prove the claim in the title by showing that a definable Grauert Direct Image Theorem in o-minimal geometry would imply a weak representability-like property of the definable Picard functor. However, this weak representability cannot hold because of the Definable Chow Theorem of Peterzil and Starchenko. v2: small typos corrected.
Cite
@article{arxiv.2601.18711,
title = {There is no Definable Grauert Direct Image Theorem},
author = {Hélène Esnault and Moritz Kerz},
journal= {arXiv preprint arXiv:2601.18711},
year = {2026}
}