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There is no Definable Grauert Direct Image Theorem

Hélène Esnault, Moritz Kerz

TL;DR

The paper proves that a universal definable Grauert Direct Image Theorem fails for non-finite morphisms in o-minimal complex analytic geometry. By constructing a smooth projective definable fibration $f: X \to S$ and a definable line bundle $\mathcal{L}$ with $f_*\mathcal{L}(n)$ not definable coherent, the authors derive a non-constant definable map $\sigma_{\mathcal{L}}: S \to \mathrm{Pic}(X/S)$, which would have to be algebraic by the Definable Chow Theorem if a definable DGDI held. The non-existence of non-constant algebraic maps from a rational variety to an abelian variety yields a contradiction, thus refuting the general definable DGDI. The work leverages Simpson-type constructions for definable sheaves, the structure of the character variety $\mathrm{Char}(Y)$, and the definable Chow theorem to delineate the limits of definable direct image theorems in this setting. It also clarifies that while a definable DGDI may hold in finite cases, it cannot extend to arbitrary projective morphisms in the current framework.

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.

There is no Definable Grauert Direct Image Theorem

TL;DR

The paper proves that a universal definable Grauert Direct Image Theorem fails for non-finite morphisms in o-minimal complex analytic geometry. By constructing a smooth projective definable fibration and a definable line bundle with not definable coherent, the authors derive a non-constant definable map , which would have to be algebraic by the Definable Chow Theorem if a definable DGDI held. The non-existence of non-constant algebraic maps from a rational variety to an abelian variety yields a contradiction, thus refuting the general definable DGDI. The work leverages Simpson-type constructions for definable sheaves, the structure of the character variety , and the definable Chow theorem to delineate the limits of definable direct image theorems in this setting. It also clarifies that while a definable DGDI may hold in finite cases, it cannot extend to arbitrary projective morphisms in the current framework.

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.
Paper Structure (5 sections, 6 theorems, 17 equations)

This paper contains 5 sections, 6 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. The Construction
  3. Definability of sections of the Picard scheme
  4. The character variety and definable local systems of rank one
  5. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

There exists a smooth projective definable morphism $f\colon X\to S$ of definable complex analytic spaces and a definable coherent sheaf $\mathcal{F}$ on $X$ such that $f_* \mathcal{F}$ is not definable coherent. More precisely, for $S=\mathbb A^1_{\mathbb C}\setminus \{ 0 \}$, $X=Y\times S$ and $f$

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Claim 2.4
  • Lemma 2.5
  • proof
  • ...and 4 more