On the irrationality of moduli spaces of projective hyperkähler manifolds

The aim of this paper is to estimate the irrationality of moduli spaces of hyperkähler manifolds of types K3${}^{[n]}$, Kum${}_n$, OG6, and OG10. We prove that the degrees of irrationality of these moduli spaces are bounded from above by a universal polynomial in the dimension and degree of the manifolds they parametrize. We also give a polynomial bound for the degrees of irrationality of moduli spaces of $(1,d)$-polarized abelian surfaces.