Regenerations and applications

TL;DR

The authors address the problem of producing uniruled divisors on projective IHS manifolds by extending the regeneration principle to the IHS setting. They prove a Regeneration principle: in a family ${\\mathcal{X}}\\to B$ with central fiber satisfying a key Hypothesis, an integral uniruled divisor on the central fiber regenerates to the total family; they verify the hypothesis for $K3^{[n]}$-type and generalized Kummer type, and deduce that polarized $(X,H)$ in these types have a uniruled divisor in some $|mH|$. They further show that, for very general moduli points, any two points can be joined by a chain of at most $2n$ rational curves, each deforming to cover a divisor, yielding density of uniruled divisors and effective non-hyperbolicity; and if ${\\rm Bir}(X)$ is infinite, then $X$ has infinitely many uniruled divisors. This work provides a unified, flexible tool to approach existence questions beyond primitive rational curves and links regeneration to Hodge loci and deformation theory.

Abstract

Chen-Gounelas-Liedtke recently introduced a powerful regeneration technique, a process opposite to specialization, to prove existence results for rational curves on projective $K3$ surfaces. We show that, for projective irreducible holomorphic symplectic manifolds, an analogous regeneration principle holds and provides a very flexible tool to prove existence of uniruled divisors, significantly improving known results.

Theorems & Definitions (20)

• Definition 1.1
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• proof : Proof of the Regeneration principle \ref{['regprin']}
• Proposition 2.1
• proof
• proof : Proof of Theorem \ref{['thm:anyreg']}
• proof : Proof of Theorem \ref{['thm:to zal']}