Parameterized family

K3^[n]-type

Parameter n ≥ 2.

Instances

K3^[2]-type

Notes

The Hilbert scheme of length-n subschemes of a K3 surface is an irreducible holomorphic symplectic variety of complex dimension 2n (Beauville 1983). Göttsche’s generating function gives its Hodge–Poincaré series (Göttsche 1994). The hyperkähler catalogue lists values for individual n (Hyperkaehler.info, n.d.).

Beauville, Arnaud. 1983. “Variétés kählériennes Dont La Première Classe de Chern Est Nulle.” Journal of Differential Geometry 18 (4): 755–82. https://doi.org/10.4310/jdg/1214438181.

Göttsche, Lothar. 1994. Hilbert Schemes of Zero-Dimensional Subschemes of Smooth Varieties. Vol. 1572. Lecture Notes in Mathematics. Springer. https://doi.org/10.1007/BFb0073491.

Hyperkaehler.info: The Geography of Compact Irreducible Holomorphic Symplectic Varieties. n.d. https://hyperkaehler.info/.