O-minimal GAGA in Definable Complex Geometry

• O-minimal GAGA is the o-minimal analog of classical Chow and GAGA theorems, asserting that definable complex analytic spaces and coherent sheaves manifest algebraicity under tameness conditions.
• It employs definabilization, coherent sheaf theory, and polynomial volume estimates to algebraize proper definable images and analytic morphisms.
• Applications include proving quasi-projectivity of period map images and linking model-theoretic, Lie-theoretic, and algebraic invariants through tameness.

Searching arXiv for recent and foundational papers on o-minimal GAGA and closely related definable-algebraization results. O-minimal GAGA is the o-minimal analog of classical Chow and GAGA theorems, asserting that “analytic equals algebraic” in the o-minimal context, provided one works in the category of complex analytic spaces and coherent sheaves that are definable with respect to a given o-minimal structure (Bakker et al., 2018). In the modern formulation, it combines a theory of definable complex analytic spaces, a GAGA-type theorem for definable coherent sheaves on algebraic spaces, algebraization of proper definable images of algebraic spaces, and affine criteria based on polynomial volume growth for definable analytic sets (Brosnan, 2021). The resulting framework has been used to prove quasi-projectivity of images of period maps, ampleness of the Griffiths bundle, and comparison statements between model-theoretic and Lie-theoretic invariants.

1. Definable analytic geometry and the o-minimal setting

An o-minimal structure on the real field specifies a family of subsets of $\mathbb{R}^n$ for all $n$ enjoying strong finiteness and tameness properties, with one-dimensional definable sets given by finite unions of points and intervals (Brosnan, 2021). In this setting, a definable complex analytic space is locally modeled on definable analytic subspaces of $\mathbb{C}^n$, with transition functions and structure sheaf given by holomorphic functions whose graphs are definable. A definable coherent sheaf is a sheaf of modules over the structure sheaf of a definable complex analytic space, locally given as cokernels of morphisms defined by matrices of definable holomorphic functions, with stalks of finite type and the usual coherence property (Bakker et al., 2018).

The basic algebraic-to-definable passage is definabilization. If $X$ is an algebraic space over $\mathbb{C}$, its definabilization $X^{\mathrm{def}}$ is a definable complex analytic space associated functorially to $X$. This construction provides the ambient category in which o-minimal GAGA is formulated: algebraic spaces and coherent sheaves are compared not with arbitrary analytic objects, but with analytic objects constrained by o-minimal definability (Bakker et al., 2018).

This restriction is the decisive difference from classical analytic geometry. Definable complex analytic spaces generalize both algebraic varieties and analytic spaces, but only the “tame” analytic objects enjoy the algebraicity properties characteristic of o-minimal GAGA. A central theme is that o-minimality supplies global finiteness phenomena—finite definable refinements, stabilization, and triangulation—that can substitute for compactness in classical GAGA arguments (Bakker et al., 2018).

2. The GAGA theorem for definable coherent sheaves

The foundational sheaf-theoretic statement is the o-minimal GAGA theorem for coherent sheaves. For an algebraic space $X$, the definabilization functor

$$(-)^{\mathrm{def}} : \mathrm{Coh}(X) \to \mathrm{Coh}(X^{\mathrm{def}})$$

is fully faithful and exact, and its essential image is closed under taking subobjects and quotients in $\mathrm{Coh}(X^{\mathrm{def}})$ (Bakker et al., 2018). In particular, algebraic coherent sheaves are completely faithfully embedded among definable coherent sheaves, and every definable coherent subsheaf or quotient of a definabilized algebraic coherent sheaf is again algebraic.

The proof strategy follows the classical analytic pattern, but within the definable category. The theory of coherent sheaves on definable complex analytic spaces is developed using covers and refinements that are finite and definable. The definable structure sheaf satisfies Oka Coherence and a Noetherian property, and definable versions of Weierstrass preparation, Weierstrass division, and the Nullstellensatz are available. Full faithfulness and algebraicity of subobjects are obtained by induction on dimension together with o-minimal Chow-type results (Bakker et al., 2018).

A significant limitation is equally explicit: essential surjectivity fails. The theory does not claim that every definable coherent sheaf is algebraic; rather, it identifies a robust algebraic core inside the definable coherent category. This point is central to the correct interpretation of o-minimal GAGA. It is not an equivalence between all definable coherent sheaves and algebraic coherent sheaves, but a precise algebraization theorem with strong closure properties under standard exact operations (Bakker et al., 2018).

3. Proper definable images and the replacement of properness by tameness

Classical GAGA, in Serre’s form, provides an equivalence between coherent algebraic and analytic sheaves for proper algebraic varieties. O-minimal GAGA changes the hypothesis: it removes the properness hypothesis provided one works in the definable analytic category (Bakker et al., 2018). The key image theorem is:

Let $n$0 be a separated algebraic space of finite type over $n$1, $n$2 a definable complex analytic space, and $n$3 a proper definable map. Then $n$4 is, uniquely up to isomorphism, the definabilization of a morphism of algebraic spaces (Bakker et al., 2018).

This statement algebraizes proper definable images of complex algebraic spaces. In the classical analytic setting, proper analytic images need not be algebraic; in the o-minimal setting, definability rules out the analytic pathologies that obstruct algebraization. The theorem is therefore not merely a sheaf-theoretic comparison, but a geometric algebraization principle for morphisms and images.

Artin’s algebraization of formal modifications enters the proof to handle nilpotent thickenings and stepwise algebraization over non-reduced strata. The role of o-minimality is to control the analytic side sufficiently strongly that algebraization becomes unique and functorial. In this sense, o-minimal GAGA is a replacement of compactness by tameness: algebraicity is recovered not from properness alone, but from properness together with definability (Bakker et al., 2018).

4. Affine forms, definable Chow, and volume growth

In affine space, o-minimal GAGA appears as a definable Chow theorem. Any closed complex analytic subset of $n$5 that is definable in an o-minimal expansion of the real field is algebraic (Brosnan, 2021). Brosnan gives a very quick proof of this using the Bishop–Stoll theorem together with a volume estimate for definable sets.

The quantitative input is the volume estimate: if $n$6 is a definable set of dimension $n$7, then

$n$8

where $n$9 denotes the $\mathbb{C}^n$0-dimensional Hausdorff measure and $\mathbb{C}^n$1 is a constant depending on $\mathbb{C}^n$2 (Brosnan, 2021). For a closed complex analytic definable set $\mathbb{C}^n$3 of complex dimension $\mathbb{C}^n$4, its definable dimension as a subset of $\mathbb{C}^n$5 is $\mathbb{C}^n$6, so the estimate yields

$\mathbb{C}^n$7

Bishop–Stoll then implies that $\mathbb{C}^n$8 is algebraic (Brosnan, 2021).

This formulation makes the mechanism of o-minimal GAGA especially transparent. Definability enforces polynomial volume growth, and polynomial volume growth is the analytic criterion that forces algebraicity. The argument shows that tame geometry at infinity is enough to collapse the distinction between analytic and algebraic subsets in affine space. It also provides a bridge from model-theoretic tameness to classical analytic criteria in the sense of Bishop, Stoll, and Lelong theory (Brosnan, 2021).

5. Period maps, Griffiths’ conjecture, and Hodge-theoretic applications

A principal application of o-minimal GAGA is the proof of Griffiths’ conjecture on the quasi-projectivity of images of period maps. For a reduced, separated algebraic space $\mathbb{C}^n$9 of finite type over $X$0 and a period map

$X$1

the image algebraizes: $X$2 factors uniquely as

$X$3

where $X$4 is a dominant map of reduced algebraic spaces and $X$5 is a closed immersion of analytic spaces (Bakker et al., 2018). Moreover, the Griffiths $X$6-bundle $X$7 restricted to $X$8 is the analytification of an ample algebraic $X$9-bundle, so $\mathbb{C}$0 is a quasi-projective variety.

The logic of the application has several layers. Period domains and period maps are definable in appropriate o-minimal structures, so their images lie in the definable analytic category. O-minimal GAGA algebraizes the relevant coherent sheaves and morphisms, while the algebraization theorem for proper definable images converts the analytic image into an algebraic space. Metric positivity then yields the ampleness statement for the Griffiths bundle (Bakker et al., 2018).

Further consequences are recorded in the same framework. If $\mathbb{C}$1 is a reduced, separated Deligne–Mumford stack of finite type over $\mathbb{C}$2 admitting a quasi-finite period map, then its coarse moduli space is quasi-projective. There is also a Borel-type algebraicity statement: analytic maps to such moduli spaces from reduced algebraic varieties are algebraic. In addition, if suitable infinitesimal period map conditions are satisfied, the lowest step of the Hodge filtration is ample (Bakker et al., 2018).

6. Broader interpretations and neighboring tame analogues

In definable group theory, the expression “o-minimal GAGA principle” is used in a broader comparison-theoretic sense. For definable groups in a saturated o-minimal expansion of a real closed field, it refers to the close compatibility between the model-theoretic world and the analytic Lie group world (Conversano et al., 2011). For definably compact groups, $\mathbb{C}$3, and $\mathbb{C}$4 corresponds to the real Lie group $\mathbb{C}$5. For general definable groups, the nontrivial quotient $\mathbb{C}$6 arises from definable central extensions of semisimple groups with no definably compact parts and is described as a quotient of a connected compact commutative Lie group by a dense finitely generated subgroup, with the universal cover of the associated real semisimple Lie group supplying the topological source of the phenomenon (Conversano et al., 2011). In this usage, o-minimal GAGA is a principle of comparison between definable and Lie-theoretic structure rather than a theorem about coherent sheaves.

Neighboring tame frameworks exhibit structures that the literature itself presents as GAGA analogues rather than full o-minimal GAGA theorems. In definably complete locally o-minimal structures satisfying the discreteness property that the image of a nonempty definable discrete set under a coordinate projection is again discrete, one has an addition property of dimension for equi-dimensional fibers,

$\mathbb{C}$7

decomposition into quasi-special submanifolds, constructibility of definable sets, and frontier conditions for the resulting partitions (Fujita, 2020). The paper explicitly presents these as “Potential GAGA Analogues”: topological and geometric invariants defined via definability behave as in classical algebraic and analytic geometry, and the results are said to lay the foundation for further “tame” or “definable GAGA” theorems.

A related perspective appears in o-minimalistic structures. There, DCTC, quasi-cell decomposition, the Grothendieck ring, the Discrete Pigeonhole Principle, and Taylor sets are used to formulate “GAGA-type results” as transfers between analytic and algebraic, or between tame and o-minimalistic, categories (Schoutens, 2011). The emphasis is not on algebraization of coherent sheaves, but on the persistence of dimension theory, monotonicity, quasi-cell decomposition, and Euler-characteristic-type invariants across contexts. This suggests that o-minimal GAGA belongs to a wider program in tame geometry: analytic objects become algebraically rigid when definability supplies sufficiently strong finiteness and stratification properties.