Finite abelian groups acting on rationally connected threefolds II: groups of K3 type
Konstantin Loginov, Antoine Pinardin, Zhijia Zhang
TL;DR
The paper classifies finite abelian subgroups G of the birational automorphism group of a complex rationally connected threefold X, showing that G is either of product type or of K3 type with a short exact sequence 0→ℤ/m→G→H→0, where H acts on a K3 surface. Using equivariant minimal model programs, K3-surface automorphism theory (notably BH23), orbifold Riemann–Roch, and lattice techniques, the authors bound m and constrain H to six maximal K3-type groups; among these, only four yield non-product-type examples. They provide explicit constructions (e.g., Fermat hypersurfaces in weighted projective spaces) illustrating K3-type actions, and they prove that in all analyzed h^0(−K_X) cases with h^0(−K_X)≥2 or =1, G must be of product type or one of the four exceptional K3-type groups. The results connect threefold birational symmetry with K3 surface automorphisms, giving a finite, largely explicit classification and offering a framework for potential complete classifications under a conjectured product-type list. The work thus advances understanding of the Cremona group Cr_3(ℂ) and the role of K3-type symmetries in rationally connected threefolds.
Abstract
We study finite abelian groups acting on three-dimensional rationally connected varieties. We concentrate on the groups of K3 type, that is, abelian extensions by a cyclic group of groups that faithfully act on a K3 surface. In particular, if a finite abelian group faithfully acts on a threefold preserving a K3 surface (with at worst du Val singularities), then such a group is of K3 type. We prove a classification theorem for the groups of K3 type which can act on three-dimensional rationally connected varieties. We note the relation between certain groups of K3 type and K3 surfaces with higher Picard number.