Definable Grauert Direct Image Theorem

• The definable Grauert direct image theorem examines the failure to extend classical finiteness and base-change properties to definable coherent sheaves under o-minimal geometry.
• It identifies a critical obstruction arising from the definable Chow theorem, which prevents the representability necessary for definable morphisms.
• Research implications include restricted moduli construction for analytic bundles and potential validity only under extra geometric conditions.

The definable Grauert direct image theorem concerns the extension of classical finiteness and base-change results for higher direct images of coherent analytic sheaves to the category of definable complex-analytic spaces in the setting of o-minimal geometry. The main result is negative: there exists no reasonable analogue of the Grauert direct image theorem for definable coherent sheaves in the o-minimal context, due to an incompatibility with the definable Chow theorem and the resulting obstruction to functor representability in this category (Esnault et al., 26 Jan 2026).

1. Classical Grauert Direct Image Theorem

The classical Grauert direct image theorem, foundational in both complex-analytic and algebraic geometry (notably through the GAGA correspondence), states: let $f\colon X \to S$ be a proper holomorphic map of complex analytic spaces and $\mathcal{F}$ a coherent analytic sheaf on $X$. Then for every $i\ge0$:

• $R^i f_*\mathcal{F}$ is a coherent $\mathcal{O}_S$-module.
• The formation of the stalk $(R^i f_*\mathcal{F})_s$ commutes with analytic base change at $s\in S$.

For $i=0$, $f_* \mathcal{F}$ is coherent. These results yield powerful finiteness and base-change properties used throughout the study of complex and algebraic spaces.

2. Formulation of a Definable Grauert Theorem in the O-minimal Setting

In o-minimal geometry, the objects of study are definable complex-analytic spaces, as developed by Peterzil–Starchenko and Bakker–Brunebarbe–Tsimerman. These are complex analytic spaces with atlases whose charts and coordinate overlaps are definable in a fixed o-minimal structure (e.g., as an expansion of $\mathbb{R}_{\mathrm{an}}$). Definable coherent sheaves are sheaves of $\mathcal{O}_U$-modules with locally finite definable presentations.

A potential "definable Grauert direct-image theorem" would, by analogy, assert for a proper definable analytic map $f\colon X \to S$ and definable coherent sheaf $\mathcal{F}$ on $X$:

• For all $i\ge0$, $R^i f_* \mathcal{F}$ is definable coherent on $S$.
• The usual base-change properties hold in the definable category.

In particular, $f_* \mathcal{F}$ would be definable coherent for $i=0$. This would extend essential finiteness and base-change mechanisms to the o-minimal/definable setting.

3. Representability and the Definable Picard Functor

Assume $f\colon X \to S$ is projective, flat, with geometrically integral fibers, and such a definable direct-image result holds. Grothendieck’s Picard scheme, $\operatorname{Pic}(X/S)$, represents the functor sending $T$ to the group of isomorphism classes of line bundles on $X_T$.

Given a definable line bundle $\mathcal{L}$ on $X_U$, the section

$$\sigma_{\mathcal{L}}: U \to \operatorname{Pic}(X/S)|_U$$

associates to $u\in U$ the class of $\mathcal{L}|_{X_u}$. If $f_*(\mathcal{L}(n))$ is definable coherent for all sufficiently large $n$, then $\sigma_{\mathcal{L}}$ is definable analytic. Thus, the definable Grauert theorem would imply a weak representability property: every definable family of line bundles gives a definable morphism into $\operatorname{Pic}(X/S)$. In other terms, the functor of definable sections is represented within the o-minimal analytic category by the Picard scheme itself.

4. The Definable Chow Theorem and Its Obstruction

Peterzil–Starchenko’s definable Chow theorem establishes that if $Y$ is a complex algebraic variety and $W \subset Y$ a definable complex-analytic subset (in the o-minimal sense), then $W$ is algebraic. Any definable analytic map between algebraic varieties is algebraic.

In particular, a definable analytic map

$$\phi: \mathbb{G}_m = \mathbb{A}^1 \setminus \{0\} \longrightarrow A$$

into an abelian variety $A$ must be algebraic (after translation), but it is classical that any algebraic map from $\mathbb{G}_m$ to an abelian variety is constant. Therefore, the existence of a non-constant definable analytic map $\mathbb{G}_m \to A$ is prohibited, directly obstructing the weak representability property required for the definable Grauert theorem.

5. Construction of the Counterexample

The obstruction is realized by a concrete counterexample following these steps:

• Character variety and definable families of local systems: For a smooth projective variety $Y$ with $H_1(Y, \mathbb{Q}) \neq 0$, its rank-one character variety $$\operatorname{Char}(Y) = \mathrm{Hom}(\pi_1^{\mathrm{ab}}(Y), \mathbb{C}^{\times})$$ is a positive-dimensional algebraic torus. Every definable analytic map $U \to \operatorname{Char}(Y)$ yields a definable analytic family of rank-one local systems on $Y \times U$.
• Line bundles and Picard scheme sections: A non-constant algebraic homomorphism $\mathbb{G}_m \to \operatorname{Char}(Y)$ gives a definable family of local systems $\mathcal{L}$ on $Y \times \mathbb{G}_m$, hence a definable analytic line bundle $\mathcal{L}_{\mathrm{an}}$ on $X = Y \times \mathbb{G}_m$. The section

$$\sigma_{\mathcal{L}}: \mathbb{G}_m \to \operatorname{Pic}(Y) \times \mathbb{G}_m \cong \operatorname{Pic}(X/\mathbb{G}_m)$$

is non-constant.

• Failure of definable coherence: If a definable Grauert theorem held, $f_*(\mathcal{L}(n))$ would be definable coherent for all large $n$, forcing $\sigma_{\mathcal{L}}$ to be definable analytic, and hence algebraic by the Chow theorem—contradicting its non-constancy.

Theorem 1.1 (Esnault–Kerz) asserts: there exists a smooth projective definable morphism $f: X \to S$ and a definable coherent sheaf $\mathcal{F}$ such that $f_* \mathcal{F}$ is not definable coherent (Esnault et al., 26 Jan 2026). Thus, the definable Grauert direct image theorem fails in general, beyond the finite case established in [BBT23, Prop. 2.52].

6. Extensions, Limitations, and Outlook

The counterexample uses only fundamental properties such as definable Oka coherence and the definable Chow theorem and applies in any o-minimal expansion of $\mathbb{R}_{\mathrm{an}}$. It is plausible that this failure holds in even broader o-minimal structures.

From the perspective of Hodge theory and vector bundle moduli on curves, the obstruction implies that moduli spaces of definable analytic bundles cannot generally be constructed via definable GAGA arguments, at least for non-finite parametrizations. A plausible implication is that definable direct-image theorems may still be accessible under additional geometric restrictions—potentially including semi-simplicity or tameness conditions.

Conversely, for finite morphisms, the definable Grauert theorem is valid (Bakker–Brunebarbe–Tsimerman). Thus, while the o-minimal-analytic category supports many analogues of classical results, the general Grauert direct image theorem’s extension is tightly blocked by the definable Chow theorem and the nontriviality of definable families of local systems.

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Key references:

• (Esnault et al., 26 Jan 2026) “There is no Definable Grauert Direct Image Theorem”
• PS09
• BBT23