Precise Reconstruction Criteria for Anabelian Varieties

Definitions and mathematical statement

The supplied record asks what it means to reconstruct an "anabelian variety" $X$ over a field $k$ from its étale fundamental group, particularly by reconstructing isomorphisms or morphisms. It does not specify a class of fields, a definition of anabelian variety, the morphisms allowed, or the arithmetic structure retained on the group. Consequently, it does not determine a single conjecture with fixed hypotheses and quantifiers.

The following conditional framework makes the reconstruction question precise without asserting that these additional choices are prescribed by the source. Fix a field $k$, a separable closure $k^{\mathrm{sep}}$, and the absolute Galois group $G_k=\operatorname{Gal}(k^{\mathrm{sep}}/k)$. For a geometrically connected $k$-variety $Z$ with a geometric base point $\bar z$ over $k^{\mathrm{sep}}$, write $\Pi_Z=\pi_1^{\acute{e}t}(Z,\bar z)$, the profinite group classifying pointed finite étale covers of $Z$. Its structural morphism gives an exact sequence \[ 1\longrightarrow\pi_1^{\acute{e}t}(Z_{k^{\mathrm{sep}}},\bar z)\longrightarrow\Pi_Z\xrightarrow{p_Z}G_k\longrightarrow1. \] Thus the arithmetic datum in this framework is the profinite group together with its projection $p_Z$, rather than merely its underlying abstract group.

For two such varieties $X,Y$, define \[ \operatorname{Hom}^{\mathrm{out}}_{G_k}(\Pi_X,\Pi_Y) =\left\{\varphi:\Pi_X\to\Pi_Y\mid\varphi\text{ is continuous and }p_Y\circ\varphi=p_X\right\}/\sim, \] where $\varphi\sim\psi$ means that $\psi=c_h\circ\varphi$ for some $h\in\ker(p_Y)$, with $c_h(g)=hgh^{-1}$. Let $\operatorname{Isom}^{\mathrm{out}}_{G_k}(\Pi_X,\Pi_Y)$ be the corresponding classes represented by topological group isomorphisms. Functoriality supplies comparison maps \[ \operatorname{Hom}_k(X,Y)\longrightarrow\operatorname{Hom}^{\mathrm{out}}_{G_k}(\Pi_X,\Pi_Y), \qquad \operatorname{Isom}_k(X,Y)\longrightarrow\operatorname{Isom}^{\mathrm{out}}_{G_k}(\Pi_X,\Pi_Y). \] Here $\operatorname{Hom}_k$ and $\operatorname{Isom}_k$ denote morphisms and isomorphisms over $k$. Setting $Y=X$ addresses endomorphisms and automorphisms of $X$; introducing a separate $Y$ describes a possible comparison formulation, not an additional source hypothesis.

Reconstruction question, with unresolved scope. Specify the intended fields, varieties, and allowable morphisms, and determine whether the appropriate comparison maps are injective, what their images are, and whether they are bijective onto a specified class of group homomorphisms or isomorphisms. Injectivity expresses uniqueness of a geometric realization, while membership in the image expresses existence. The supplied record does not choose among these targets or specify restrictions on the group homomorphisms. It also does not say whether the datum is $\Pi_X\to G_k$, the bare profinite group, or a richer étale homotopy type. These alternatives must remain unresolved rather than being combined into an asserted general reconstruction theorem.

Short English statement

The question asks which geometric maps of an anabelian variety can be recovered, and recovered uniquely, from its étale fundamental group. A standard arithmetic formulation compares maps over $k$ with continuous group maps compatible with $G_k$, modulo geometric conjugation. The supplied record leaves the fields, varieties, allowed maps, and exact reconstruction criterion unspecified, so this is a conditional framework requiring human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete source record for permanent ID $30003766$, title Precise Reconstruction Criteria for Anabelian Varieties, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2018.
• Supplied source citation: Mini-Workshop: Arithmetic Geometry and Symmetries around Galois and Fundamental Groups (2019), Oberwolfach Reports, pp. 1021--1080, DOI 10.4171/owr/2018/17. https://doi.org/10.4171/owr/2018/17. The supplied citation uses 2019, whereas the DOI, report label, proposed year, and year tag refer to 2018; this discrepancy has not been resolved.
• Supplied literature reference: Schmidt--Stix, "anabelian varieties." https://doi.org/10.4007/annals.2016.184.3.5. No theorem number or passage identifying the intended reconstruction statement is supplied.
• Supplied literature reference: "OWR 2018 report." https://doi.org/10.4171/owr/2018/17.
• Literature assessment dated exactly 2026-08-21: the record labels the problem partially_solved and reports strong results for hyperbolic curves and higher-dimensional Artin neighborhoods, formulations using fundamental groups and étale homotopy types, and the absence of a complete general reconstruction theorem. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying verification note says that mathematical content was preserved while notation, typography, and omitted definitions were repaired. This is a source-record qualification, not independent mathematical verification of this draft.