Quasi-Projective Fundamental Groups of Graph Links

Definitions and mathematical statement

Let $M$ be a compact oriented integral homology three-sphere, so $H_*(M;{\mathbb{Z}})\cong H_*(S^3;{\mathbb{Z}})$, and let $L\subset M$ be a link. Its exterior and link group are \[ X_L=M\setminus\operatorname{int}\nu(L),\qquad G_L=\pi_1(X_L), \] where $\nu(L)$ is a union of disjoint tubular neighborhoods of the components. The exterior and the open complement have the same fundamental group. A graph link, as defined by the source, is one whose exterior has only Seifert-fibered geometric pieces; the decomposition and any irreducibility conventions are those intended in the report.

A group $\Gamma$ is quasi-projective if there exists a connected smooth complex quasi-projective algebraic variety $Y$ with $\Gamma\cong\pi_1(Y)$. The source calls this property global and states the equivalence with quasi-projectivity. It does not require $Y$ to be projective, require $X_L$ itself to be algebraic, or require a specified dimension of $Y$.

The problem asks to determine precisely which non-Seifert graph links satisfy \[ \exists\,Y\text{ connected smooth complex quasi-projective},\qquad \pi_1(X_L)\cong\pi_1(Y). \] Seifert links are excluded from the requested classification. Under the customary interpretation these are links compatible with a Seifert fibration, and their exteriors are Seifert-fibered; the exact report convention, including exceptional exteriors, needs confirmation. Having every geometric piece Seifert-fibered is not the same condition as requiring the whole exterior to have one Seifert fibration.

The supplied assessment dated 2026-08-21 reports broad-family results from splice diagrams and Alexander polynomials, including non-globality for many non-Seifert graph links, but says the unrestricted classification is incomplete. It gives no universal obstruction, complete splice-diagram criterion, or list that could replace the full question.

Short English statement

Among graph links outside the Seifert-link case, which exterior groups can occur as fundamental groups of smooth complex quasi-projective varieties? The question concerns the abstract group and asks for a full characterization, beyond the supplied partial-family results.

Sources

• Local UnsolvedMath registry, permanent ID $30000823$, exact source code catalogue entry, proposed year 2007; no individual proposer is supplied.
• Mini-Workshop: Topology of closed one-forms and Cohomology Jumping Loci (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/40, pp. 2321--2360. https://doi.org/10.4171/owr/2007/40.
• Supplied reference labelled "graph-manifold characteristic varieties": https://arxiv.org/abs/1904.00729.
• The supplied corrected_verified note reports checking mathematical content against the official report. The partially solved label and literature assessment checked 2026-08-21 are dated supplied triage requiring independent verification of status and cited theorem scope.