# Hyperkaehler.info

the geography of compact irreducible holomorphic symplectic (or hyperkähler) varieties

### Hodge diamond

The Hodge diamond is a convenient way of encoding the Hodge numbers $\mathrm{h}^{p,q}=\dim_{\mathbb{C}}\operatorname{H}^q(X,\Omega_X^p)$. For hyperkähler varieties there are three symmetries:

• Hodge symmetry, which gives $\mathrm{h}^{p,q}(X)=\mathrm{h}^{q,p}(X)$, which is reflection along the vertical axis
• Serre duality, which gives $\mathrm{h}^{p,q}(X)=\mathrm{h}^{2n-p,2n-q}(X)$, which is point reflection through the middle
• an additional hyperkähler symmetry, which gives $\mathrm{h}^{p,q}(X)=\mathrm{h}^{2n-p,q}(X)$, which is reflection along the horizontal axis

In total the symmetries are given by the dihedral group of order 8 (instead of the Klein group of order 4).

K3[n]-type
The generating function (for an arbitrary surface $S$) can be found in [Theorem 2.3.14, MR1312161]. $\sum_{n=1}^{+\infty}\mathrm{h}(S^{[n]},x,y)t^n = \prod_{k=1}^{+\infty}\prod_{p,q=0}^2\left( 1 + (-1)^{p+q+1}x^{p+k-1}y^{q+k-1}t^k \right)^{(-1)^{p+q+1}\mathrm{h}^{p,q}(S)}$
Kumn-type
A closed expression can be found in [Corollary 1, MR1219901], by cancelling the Hodge polynomial of the abelian surface in $\mathrm{h}(A\times\mathrm{K}^n(A),-x,-y) = \sum_{\alpha\in\mathrm{P}(n)} \left( g(\alpha)^4 (xy)^{n-|\alpha|} \prod_{i,\alpha_i\neq 0} \left( \sum_{\beta\in\mathrm{P}(\alpha_i)} \prod_{j} \frac{1}{j^{\beta_j}\beta_j!} \left( (1-x^j)(1-y^j) \right)^{2\beta_j} \right) \right)$
OG6
The Hodge numbers are computed in [Theorem 1.1, MR3798592].
OG10
The Hodge numbers are computed in [Theorem A, 1905.03217].

##### References
MR1312161
Göttsche, Lothar. "Hilbert schemes of zero-dimensional subschemes of smooth varieties." Lecture Notes in Mathematics, 1572. doi:10.1007/BFb0073491
MR1219901
Göttsche, Lothar and Soergel, Wolfgang. "Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces." In: Math. Ann. 296 (1993), pp. 235–245. doi:10.1007/BF01445104
MR3798592
Mongardi, Giovanni and Rapagnetta, Antonio and Saccà, Giulia. "The Hodge diamond of O'Grady's six-dimensional example." In: Compos. Math. 154 (2018), pp. 984–1013. doi:10.1112/S0010437X1700803X
1905.03217
de Cataldo, Mark Andrea A. and Rapagnetta, Antonio and Saccà, Giulia. "The Hodge numbers of O'Grady 10 via Ngô strings". arXiv:1905.03217