On the Zariski density of rational curves on IHS manifolds

TL;DR

This paper demonstrates the existence of infinitely many ruled divisors on certain projective irreducible holomorphic symplectic manifolds, extending techniques from K3 surfaces to higher dimensions.

Contribution

It generalizes the controlled degeneration technique to higher dimensions and proves the existence of ruled divisors on a broad class of IHS manifolds.

Findings

Existence of infinitely many ruled divisors on IHS manifolds of K3^{[n]} or generalized Kummer type.

Extension of degeneration techniques to higher dimensions.

Application of the regeneration principle in the context of IHS manifolds.

Abstract

In analogy with recent works on $K3$ surfaces, we study the existence of infinitely many ruled divisors on projective irreducible holomorphic symplectic (IHS) manifolds. We prove such an existence result for any projective IHS manifold of $K3^{[n]}$ or generalized Kummer type which is not a variety defined over $\overline{\mathbb{Q}}$ with Picard number one or maximal. The result is obtained as a combination of the regeneration principle and of a generalization to higher dimension of a controlled degeneration technique, invented by Chen, Gounelas and Liedtke in dimension 2.