There is no Definable Grauert Direct Image Theorem

TL;DR

The paper demonstrates that a definable Grauert Direct Image Theorem cannot exist in o-minimal geometry because it would imply a property that contradicts the Definable Chow Theorem, highlighting limitations in definable complex geometry.

Contribution

It shows the non-existence of a definable Grauert Direct Image Theorem in o-minimal geometry by linking it to the non-representability of the definable Picard functor.

Findings

A definable Grauert Direct Image Theorem would imply a weak representability of the definable Picard functor.

Such weak representability is impossible due to the Definable Chow Theorem.

The result clarifies limitations of definable complex geometry in o-minimal structures.

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.