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

Beauville–Fujiki form

The Beauville–Fujiki form is an integral indivisible quadratic form on $\mathrm{H}^2(X,\mathbb{Z})$.

If we write \begin{equation} \mathrm{q}_X\colon\operatorname{H}^2(X,\mathbb{C})\to\mathbb{C} \end{equation} then there exists some constant $\mathrm{c}_X\in\mathbb{Q}_+$ (the Fujiki constant) such that on a $2n$-dimensional hyperkähler manifold $X$ we have \begin{equation} \int_X\alpha^{2n}=\mathrm{c}_X\frac{(2n)!}{n!2^n}\mathrm{q}_X(\alpha)^n \end{equation} for $\alpha\in\mathrm{H}^2(X,\mathbb{C})$.

It has index $(3,\mathrm{b}_2(X)-3)$ (when restricted to $\mathrm{H}^2(X,\mathbb{R})$).

Its discriminant is the finite abelian group $\Lambda^\vee/\Lambda$ where $\Lambda=(\operatorname{H}^2(X,\mathbb{Z}),\mathrm{q}_X)$ is the lattice. It is zero for a unimodular lattice (such as $\mathrm{E}_8$) and the hyperbolic plane $\mathrm{U}$. Because in all known cases the Beauville–Fujiki form $\Lambda$ is of the form $\Lambda'\oplus\mathrm{U}^{\oplus 3}\oplus\Lambda''$ with $\Lambda'$ unimodular and $\Lambda''$ of rank 1 or 2, the discriminant is determined by $\Lambda''$.


The Beauville–Fujiki form generalises the intersection form on a K3 surface (where $\mathrm{c}_X=1$), and more generally equips the second cohomology of a hyperkähler manifold with a lattice structure that governs much of its geometry.

type Beauville–Fujiki lattice index Fujiki constant $\mathrm{c}_X$ discriminant
K3 $\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^{\oplus3}$ $(3,19)$ 1 1
K3[n]-type $\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^{\oplus3}\oplus(-2(n-1))$ $(3,20)$ 1 $\mathbb{Z}/(2n-2)\mathbb{Z}$
Kumn-type $\mathrm{H}^{\oplus3}\oplus(-2(n+1))$ $(3,4)$ $n+1$ $\mathbb{Z}/(2n+2)\mathbb{Z}$
OG6 $\mathrm{U}^{\oplus3}\oplus(-2)^{\oplus2}$ $(3,5)$ 4 $(\mathbb{Z}/2\mathbb{Z})^{\oplus2}$
OG10 $\mathrm{E}_8(-1)^{\oplus2}\oplus\mathrm{U}^{\oplus3}\oplus\mathrm{A}_2(-1)$ $(3,21)$ 1 $\mathbb{Z}/3\mathbb{Z}$