Definable Picard Functor in O-Minimal Geometry

• The definable Picard functor is a contravariant tool that parametrizes definable line bundles over complex-analytic spaces in an o-minimal setting.
• It mirrors the classical Picard scheme through techniques involving definable Grauert direct images, yet such methods face coherence and representability challenges.
• Obstructions, including the failure of the definable Grauert Direct Image Theorem and definable Chow constraints, highlight critical differences between classical and o-minimal moduli theories.

The definable Picard functor is a fundamental construct in the intersection of o-minimal geometry and complex-analytic geometry, designed to parametrize definable line bundles in families of definable complex-analytic spaces. While the classical Picard functor, central in algebraic geometry, is often representable by a scheme—the Picard scheme—the definable Picard functor is subject to distinct obstructions and representability issues when formulated over o-minimal structures, notably in expansions of the real field such as $\mathbb{R}_{\text{an}}$ (Kleiman, 2014, Esnault et al., 26 Jan 2026). The theoretical landscape is sharply influenced by negative results concerning direct image theorems and the algebraicity constraints arising from the definable Chow Theorem.

1. Definition of the Definable Picard Functor

Given a proper, definable complex-analytic morphism $f: X \to S$ between definable complex-analytic spaces, the definable Picard functor is the contravariant functor

$$\underline{\mathrm{Pic}}^{\mathrm{def}}_{X/S} : (\mathrm{Def}/S)^{\mathrm{op}} \to \mathrm{Groups}$$

which assigns to each definable $T \to S$ the group of isomorphism classes of definable line bundles $\mathcal{L}$ on $X \times_S T$. Here a definable line bundle denotes a locally free rank-one $\mathcal{O}$-module on $X_T$, trivialized by a finite definable open cover. For $T = S$ this specializes to the group of all definable line bundles on $X$, and for a point $s \in S$, one recovers the analytic Picard group of the fiber $X_s$: $$\underline{\mathrm{Pic}}^{\mathrm{def}}_{X/S}(\{s\}) \cong \mathrm{Pic}^{\mathrm{an}}(X_s).$$ This functor is modeled in direct analogy with the algebraic and analytic Picard functors, but formulated over the category of definable complex-analytic spaces in a fixed o-minimal expansion (Esnault et al., 26 Jan 2026).

2. Representability and the Role of Definable Grauert Direct Image

In the classical algebraic or complex-analytic settings, the Picard functor is closely linked to representability. For example, under suitable properness and flatness hypotheses, Grothendieck’s theorem asserts that the Picard functor is represented by the Picard scheme, a separated $S$-scheme locally of finite type, compatible with base change (Kleiman, 2014).

Analogously, in the definable context, the possibility of such representability—or even a weaker version thereof—depends crucially on the existence of a definable Grauert Direct Image Theorem. Specifically, if direct images $R^i f_* \mathcal{F}$ of definable coherent sheaves were again definably coherent, one could emulate Simpson's construction to produce a universal definable analytic parameter space for definable line bundles, and the functor $\underline{\mathrm{Pic}}^{\mathrm{def}}_{X/S}$ would be weakly representable by the definable space underlying the analytic Picard scheme.

This principle provides:

• A “weak representability” criterion: $\underline{\mathrm{Pic}}^{\mathrm{def}}_{X/S}$ is represented by the definable analytic space underlying $\mathrm{Pic}(X/S)$ provided definable coherence of direct images holds for all $R^i f_*\mathcal{L}(n)$.
• A direct link to moduli theory: universal parameter spaces for definable line bundles would exist at least in the analytic category and be compatible with definable structures.

3. The Impossibility of General Definable Representability

The core negative result, as proven by Esnault–Kerz, is a definitive obstruction: the definable Grauert Direct Image Theorem fails in full generality, and hence the definable Picard functor is not weakly representable outside very restricted cases (Esnault et al., 26 Jan 2026). Explicitly, there exist:

• Smooth projective definable morphisms $f: X \to S$ (with $S = \mathbb{G}_m$) and
• Definable line bundles $\mathcal{L}$ on $X$ such that, for infinitely many $n \gg 0$, the direct image sheaves $f_*(\mathcal{L}(n))$ are not definably coherent on $S$.

Therefore, there can be no universal definable analytic parameter space for definable line bundles on $X$: $\underline{\mathrm{Pic}}^{\mathrm{def}}_{X/S}$ cannot be realized by any definable analytic space in general. This sharply contrasts with the classical representability theorems for the Picard functor (Kleiman, 2014).

4. Outline of the Obstruction via the Definable Chow Theorem

The contradiction proof hinges on several steps:

• Construction: Given a smooth projective complex variety $Y$ with $H^1(Y;\mathbb{Q}) \neq 0$, its character variety $$\mathrm{Char}(Y) = \mathrm{Hom}(\pi_1^{\mathrm{ab}}(Y), \mathbb{C}^\times)$$ is positive-dimensional. A non-constant algebraic homomorphism $\mathbb{G}_m \to \mathrm{Char}(Y)$ induces a definable family of local systems, and thus definable line bundles on $X = Y \times S$.
• Non-constancy: The analytic classifying map $$\sigma_{\mathcal{L}}: S \to \mathrm{Pic}(Y) \cong \mathrm{Pic}(X/S)$$ is non-constant on the unit circle, as distinct unitary characters yield non-isomorphic Hodge-theoretic bundles.
• Definable coherence contradiction: If direct images $f_*(\mathcal{L}(n))$ were definably coherent for all large $n$, Simpson's analytic construction would render $\sigma_{\mathcal{L}}$ definable analytic.
• Definable Chow Theorem: By Peterzil–Starchenko, any definable holomorphic map from a rational variety ($\mathbb{G}_m$) to an abelian variety ($\mathrm{Pic}^0(Y) \subset \mathrm{Pic}(Y)$) is algebraic, but no non-constant such algebraic map exists (Milne, Corollary 3.9). This contradicts the earlier non-constancy.

The only failing assumption is definable coherence of direct images, establishing the negative representability result.

5. Comparison with the Classical Picard Functor and Scheme

For context, classically the Picard functor for a morphism of schemes $f: X \to S$ is first defined as the presheaf $$\mathrm{Pic}^{\text{nl}}(X/S)(T) := \mathrm{Pic}(X_T)/\operatorname{im}[\mathrm{Pic}(T) \to \mathrm{Pic}(X_T)]$$, with alternative, cohomological descriptions via

$$\mathrm{Pic}^{\text{nl}}(X/S)(T) \cong H^0(T, R^1 f_{T*} \mathbb{G}_{m, X_T}),$$

where $\mathbb{G}_m$ is the sheaf of invertible functions and $R^1 f_{T*}$ denotes the higher direct image. Under properness, flatness, and suitable cohomological flatness, the fpqc-sheafification is represented by the Picard scheme $\mathrm{Pic}_{X/S}$, which is separated, locally of finite type, compatible with base change, and provides a universal parameter space with group-scheme structure, tangent space $H^1(X_s, \mathcal{O}_{X_s})$ at the identity, and smoothness in characteristic $0$ (Kleiman, 2014).

6. Implications, Special Cases, and Open Problems

Several implications and directions arise from these results:

• There is no general o-minimal GAGA theorem in the Grauert style when $f$ has positive-dimensional fibers.
• The moduli-theoretic interpretability of the functor of definable line bundles fails even at the weak, functorial section level.
• For finite maps or families with trivial $H^1$, definable Grauert direct image may still succeed, suggesting loci of positive behavior within broader pathologies.
• Open questions include the precise classification of those morphisms $f$ for which all higher direct images of definable coherent sheaves are definably coherent, and the possibility of weaker moduli objects such as “definable Néron models” for $\mathrm{Pic}^0_{X/S}$ under supplementary hypotheses (Esnault et al., 26 Jan 2026).

7. Summary Table: Picard Functor Variants

Setting	Functor Definition	Representability
Classical (Sch/S)	$\mathrm{Pic}^{\text{nl}}(X/S)(T)$, or $H^0(T,\dots)$	Representable by the Picard scheme under standard flatness/coherency
O-minimal/Definable	$\underline{\mathrm{Pic}}^{\mathrm{def}}_{X/S}(T)$	Not representable, nor weakly representable, in general

The definable Picard functor formalizes the notion of parameterizing definable line bundles in o-minimal settings. While its functorial properties mirror those of its algebro-geometric and complex-analytic antecedents, its failure of representability is a reflection of deeper obstructions in o-minimal and model-theoretic geometry, sharply contrasting with the established theory of the Picard scheme.