K3^[2]-type

Hodge diamond

The entries are h^p,q = dim_ℂ H^q(X, Ω^p_X). Row k has p + q = k, with p increasing from left to right.

Symmetry group: D4.

Numerical invariants

Euler characteristic: 324.

Dimension of the local deformation space: 21.

Betti numbers: b₀ = 1, b₁ = 0, b₂ = 23, b₃ = 0, b₄ = 276, b₅ = 0, b₆ = 23, b₇ = 0, b₈ = 1.

Chern numbers

• ∫c₂c₂ = 828
• ∫c₄ = 324

Cohomology lattices

• H²(X; ℤ)/tors, with the Beauville–Bogomolov–Fujiki form: U^3 + E8(-1)^2 + <-2>.

Notes

The Hodge series is the n = 2 case of Göttsche’s formula (Göttsche 1994). The lattice and numerical invariants agree with the K3¹ entry of the hyperkähler catalogue (Hyperkaehler.info, n.d.).