Galois Actions on Moduli of General-Type Surfaces

Definitions and mathematical statement

Fix an embedding $\overline{{\mathbb{Q}}}\hookrightarrow{\mathbb{C}}$ and let $\Gamma=\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$. For a variety $X$ over $\overline{{\mathbb{Q}}}$, let $X^\sigma$ be its conjugate by $\sigma\in\Gamma$. Compare topological invariants of the associated complex analytic spaces through the chosen embedding.

The source asks three questions:

1. Which topological invariants of a variety over $\overline{{\mathbb{Q}}}$ are preserved by every Galois conjugation? For a proposed invariant $I$, the relevant condition is $I(X)=I(X^\sigma)$ for all varieties and conjugations in the intended class. The wording says variety generally, while the surrounding context concerns surfaces; the precise scope is not stated.
2. Is the $\Gamma$-action faithful on the union $\mathcal M$ of moduli spaces of minimal complex surfaces of general type?
3. Is it faithful on the rigid surfaces, described in the original as the zero-dimensional components?

If $\mathcal M$ means the set of geometric moduli points represented by such surfaces over $\overline{{\mathbb{Q}}}$, the faithfulness question has the precise pattern \[ \forall\sigma\in\Gamma\setminus\{1\}\ \exists[S]\in\mathcal M: [S^\sigma]\ne[S]. \] The third question restricts the witness $[S]$ to the source's rigid subset. The witness may depend on $\sigma$; the question does not say that every nontrivial element moves every surface. Faithfulness on a union is also not faithfulness on each component or for every fixed set of numerical invariants.

The record does not identify whether $\mathcal M$ denotes moduli points, connected/irreducible components, a coarse space, or another object. An absolute Galois automorphism acts directly on $\overline{{\mathbb{Q}}}$-points; an action on all complex points requires further conventions and is not supplied by arbitrary extensions to ${\mathbb{C}}$. Rigid as a zero-dimensional component is not silently strengthened to vanishing of every infinitesimal deformation group. The precise smooth/projective versus canonical-model surface class and moduli equivalence must likewise be recovered. The first question supplies no fixed list or criterion of topological invariants.

Short English statement

The entry asks which topology survives Galois conjugation and whether the Galois action is faithful on the whole general-type-surface moduli collection or already on rigid surfaces. The exact moduli object and variety class need confirmation; the three tasks remain distinct.

Sources

• Local UnsolvedMath registry, permanent ID 30000852, code catalogue entry, "Galois Actions on Moduli of General-Type Surfaces." Proposed year 2007; proposer unspecified.
• Komplexe Algebraische Geometrie (2008), Oberwolfach Reports, pp. 2791--2850, DOI 10.4171/owr/2007/47; https://doi.org/10.4171/owr/2007/47. The displayed report year 2008 differs from the DOI and proposed year 2007. This is the sole supplied literature reference.
• The original explicitly identifies the rigid-surface subset with zero-dimensional components; this qualifier is retained.
• Statement status corrected_verified; the supplied note says official-report content was checked and wording/notation normalized without changing the source claim.
• Literature triage checked 2026-08-21: open. It calls the three questions broad, reports only partial invariance results in the report, and says no complete faithfulness answer for the specified moduli set or rigid subset was verified in the supplied check. Exact moduli scope and current status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.