Classification of fundamental groups of smooth complex projective varieties

Determine which groups can occur as fundamental groups of compact Kähler manifolds or smooth complex projective varieties, with the goal of obtaining a complete classification.

Background

The paper begins by placing its realization problem in the broader context of determining which groups arise as fundamental groups of geometrically restricted spaces. It notes that fundamental groups of compact Kähler manifolds and smooth complex projective varieties satisfy strong geometric and representation-theoretic constraints, but that a complete classification remains unresolved. This is distinct from the finite-group case, for which the paper recalls a theorem realizing every finite group as the fundamental group of a smooth connected complex projective variety of any prescribed positive dimension.

References

This question has spawned a lot of work and one can show that these groups satisfy strong geometric and representation-theoretic properties, but a complete classification is still out of reach (,).

— Finite group schemes as fundamental group schemes of smooth projective varieties  (2608.27246 - Bassan, 27 Aug 2026) in Introduction

The general problem for lattices in simply connected nilpotent Lie groups remains open.

— mixed Hodge structures on the strictly upper triangular Lie algebras  (2610.23505 - Shimoji, 16 Oct 2026) in Section 1.3.1, page 3