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Beauville–Bogomolov–Fujiki form

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

If we write \[ \mathrm{q}_X\colon\operatorname{H}^2(X,\mathbb{C})\to\mathbb{C} \] 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 \[ \int_X\alpha^{2n}=\mathrm{c}_X\frac{(2n)!}{n!2^n}\mathrm{q}_X(\alpha)^n \] 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–Bogomolov–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 divisors of its exponent are exactly the possible divisibilities of primitive classes in $\operatorname{H}^2(X,\mathbb{Z})$.

The Beauville–Bogomolov–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.