On the irrationality of moduli spaces of projective hyperk\"ahler manifolds

TL;DR

This paper estimates the irrationality of moduli spaces of various hyperk"ahler manifolds and polarized abelian surfaces, providing universal polynomial bounds based on their dimensions and degrees.

Contribution

It establishes universal polynomial bounds for the degrees of irrationality of moduli spaces of hyperk"ahler manifolds and polarized abelian surfaces, advancing understanding of their geometric complexity.

Findings

Degrees of irrationality are bounded by a universal polynomial in dimension and degree.

Polynomial bounds are provided for moduli spaces of hyperk"ahler manifolds and polarized abelian surfaces.

The bounds apply to specific types of hyperk"ahler manifolds and polarized abelian surfaces.

Abstract

The aim of this paper is to estimate the irrationality of moduli spaces of hyperk\"ahler 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.