Reid’s Fundamental-Group Conjecture for General-Type Surfaces

Definitions and mathematical statement

Let $S$ be a minimal complex surface of general type, understood in the standard smooth connected projective setting: $\kappa(S)=2$ and $S$ has no exceptional curve of self-intersection $-1$. Write $K_S$ for its canonical divisor, $K_S^2$ for its self-intersection, and \[\chi(S)=\chi(\mathcal O_S)=\sum_{i=0}^2(-1)^i\dim_{\mathbb{C}} H^i(S,\mathcal O_S).\] Thus $\chi$ is the holomorphic Euler characteristic, not the topological one. Let $\pi_1^{\mathrm{alg}}(S)$ be the profinite algebraic fundamental group classifying finite étale covers. An irregular étale cover means a connected finite étale cover $T\to S$ with $q(T)=\dim_{\mathbb{C}} H^1(T,\mathcal O_T)>0$; absence of such covers includes $q(S)=0$.

Miles Reid's conjecture in the source asks whether \[K_S^2<4\chi(S)\] implies that $\pi_1^{\mathrm{alg}}(S)$ is, up to finite group extensions, the fundamental group of a curve. To compare profinite groups, the curve group must be read as an algebraic fundamental group, or as the profinite completion of the topological fundamental group in the compact complex-curve setting. The phrase up to finite group extensions does not specify the finite-kernel/finite-index diagram. For example, passing to an open subgroup and then quotienting by a finite normal subgroup is one possible convention, but it is not asserted as the report's exact formulation without further evidence. The curve class and allowed genus are also unstated; genus zero would allow the trivial curve group and hence finite cases.

The second question asks for explicit upper bounds on \[|\pi_1^{\mathrm{alg}}(S)|\] under the stronger inequality $K_S^2<3\chi(S)$ and the assumption that $S$ has no irregular finite étale cover. No candidate numerical bound or permitted dependence on the surface's invariants is supplied. The 2026-08-21 triage labels the record open, reports the conjecture proved in the irregular and $K_S^2<3\chi(S)$ cases, and locates the unresolved no-irregular-cover range at $3\chi(S)\leq K_S^2<4\chi(S)$ within the conjecture's strict upper threshold. These are supplied literature qualifications, not a newly verified classification or bound.

Short English statement

For minimal general-type surfaces below the slope-four threshold, ask whether the algebraic fundamental group is a curve group up to finite extensions. Below slope three, with no irregular finite étale cover, ask for explicit bounds on its order. The profinite interpretation and finite-extension convention require clarification.

Sources

• Local UnsolvedMath registry, permanent ID 30000854, source problem catalogue entry, Reid's Fundamental-Group Conjecture for General-Type Surfaces, proposed year 2007. The original attributes the conjecture to Miles Reid; no separate proposer metadata is supplied.
• Supplied citation: Komplexe Algebraische Geometrie (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/47, pp. 2791--2850: https://doi.org/10.4171/owr/2007/47. Supplied literature label: original Oberwolfach report.
• Supplied literature label: Mendes Lopes--Pardini, algebraic fundamental groups, https://arxiv.org/abs/math/0512483.