Constructing Highly Symmetric Compact Manifolds and Algebraic Varieties
Dávid R. Szabó
TL;DR
The paper resolves the finite-subgroup structure of birational and diffeomorphism groups by proving that, for any r and algebraically closed field k, there exist a variety X_{r,k} and a compact manifold M_r whose birational/diffeomorphism groups contain every finite nilpotent group of class at most 2 and rank at most r (order coprime to char(k) for the algebraic case). The core method combines group-theoretic reductions to Heisenberg groups, Mumford theta groups, and Appell– Humbert theory for line bundles on abelian varieties or complex tori, with a uniformisation strategy to realize a fixed base space independent of the specific group. Number theory and K-theory are employed to achieve trivial Chern characters and to ensure actions descend to compact spaces, enabling a faithful embedding on a fixed, bounded-dimensional platform. The results provide sharp characteristic-0 bounds and answer questions of Mundet i Riera regarding the universality and sharpness of such group actions in the birational and smooth settings, with broad implications for understanding Jordan-type properties in geometry.
Abstract
For every algebraically closed field $k$ and natural number $r$, we construct several algebraic varieties (over $k$) whose birational automorphism group contains every finite nilpotent group of class at most $2$, rank at most $r$ whose order is coprime to the characteristic of $k$. This construction is sharp in characteristic $0$, i.e. up to bounded extension, the set of groups from the statement cannot be replaced by a larger one. Using similar main ideas (with different technical details), for every $r$, we construct several compact manifolds whose diffeomorphism groups contain every finite nilpotent group of class at most $2$, rank at most $r$. This result answers a question of Mundet~i~Riera affirmatively and is conjecturally sharp up to bounded extension.