On the Zariski Density of Rational Curves on Irreducible Holomorphic Symplectic 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.