Divisibility
The divisibility of a primitive class $\lambda\in\operatorname{H}^2(X,\mathbb{Z})$ is the positive integer $\operatorname{div}(\lambda)$ generating the ideal \[ (\lambda,\operatorname{H}^2(X,\mathbb{Z})) = \operatorname{div}(\lambda)\mathbb{Z} \subseteq \mathbb{Z}, \] where the pairing is the Beauville–Bogomolov–Fujiki form. Together with the square $\mathrm{q}_X(\lambda)$ it is the basic numerical invariant of a class: both are invariant under the monodromy group, and the pairs $(\mathrm{q}_X(\lambda),\operatorname{div}(\lambda))$ are used to enumerate the polarised deformation types of hyperkähler varieties, see [MR2964480].
The class $\lambda/\operatorname{div}(\lambda)$ lies in the dual lattice, and defines an element of order $\operatorname{div}(\lambda)$ in the discriminant group of $\operatorname{H}^2(X,\mathbb{Z})$, so that the divisibility divides the exponent of the discriminant group. Conversely, every divisor of the exponent is the divisibility of some primitive class, because $\operatorname{H}^2(X,\mathbb{Z})$ contains a hyperbolic plane as a direct summand. This gives the following possibilities.
| type | discriminant group | possible divisibilities |
|---|---|---|
| K3 | $0$ | $1$ |
| K3[n]-type | $\mathbb{Z}/(2n-2)\mathbb{Z}$ | divisors of $2n-2$ |
| Kumn-type | $\mathbb{Z}/(2n+2)\mathbb{Z}$ | divisors of $2n+2$ |
| OG6 | $(\mathbb{Z}/2\mathbb{Z})^{\oplus2}$ | $1,2$ |
| OG10 | $\mathbb{Z}/3\mathbb{Z}$ | $1,3$ |
Which pairs $(\mathrm{q}_X(\lambda),\operatorname{div}(\lambda))$ are realised, and how they classify monodromy orbits of primitive classes, is a more refined question, see [MR2964480].
References
- MR2964480
- Markman, E. (2011). A survey of Torelli and monodromy results for holomorphic-symplectic varieties. In Complex and differential geometry (Vol. 8, pp. 257–322). Springer, Heidelberg. doi:10.1007/978-3-642-20300-8\_15 MR2964480