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, MR4338453].
References
- MR1312161
- Göttsche, L. (1994). Hilbert schemes of zero-dimensional subschemes of smooth varieties (Vol. 1572, p. x+196). Springer-Verlag, Berlin. doi:10.1007/BFb0073491 MR1312161
- MR1219901
- Göttsche, L., & Soergel, W. (1993). Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces. Math. Ann., 296(2), 235–245. doi:10.1007/BF01445104 MR1219901
- MR3798592
- Mongardi, G., Rapagnetta, A., & Saccà, G. (2018). The Hodge diamond of O’Grady’s six-dimensional example. Compos. Math., 154(5), 984–1013. doi:10.1112/S0010437X1700803X MR3798592
- MR4338453
- de Cataldo, M. A. A., Rapagnetta, A., & Saccà, G. (2021). The Hodge numbers of O’Grady 10 via Ngô strings. J. Math. Pures Appl. (9), 156, 125–178. doi:10.1016/j.matpur.2021.10.004 MR4338453