On the Zariski density of rational curves on IHS manifolds

TL;DR

The paper addresses the problem of generating abundant ruled divisors and, by extension, rational curves on projective irreducible holomorphic symplectic (IHS) manifolds. It develops a higher-dimensional regeneration framework and a controlled-degeneration principle that propagates ruled divisors from very general members of moduli to all fibers within a component, specifically for IHS manifolds of $K3^{[n]}$ or generalized Kummer type. The main results establish the existence of infinitely many ample ruled divisors for all manifolds in a moduli component with $2\le \rho(X)<\rho_{\max}$, and also for manifolds not defined over $\overline{\mathbb{Q}}$, along with weak-to-strong propagations of conjectures on rational curves. The work further shows how to transfer these geometric phenomena through domination and discusses consequences for density in moduli, lattice-theoretic conditions, and certain OG6-type varieties, highlighting a robust mechanism to produce Zariski-dense families of rational curves in IHS geometry.

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.

Theorems & Definitions (35)

• Conjecture 1.1: cf. Huy16, Chapter 13, Conjecture 0.2
• Conjecture 1.2
• Conjecture 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Proposition 2.1: CGL, Proposition 7.4