On anti-hyperbolicity for hyperkähler varieties

TL;DR

The paper advances the study of anti-hyperbolicity for hyperkähler varieties by establishing dominability criteria and the existence of dense entire curves. It generalizes prior two-dimensional results to higher-dimensional hyperkähler manifolds, showing that dual Lagrangian fibrations yield meromorphic dominability by $\mathbb{C}^{2n}$, while a single Lagrangian fibration over $\mathbb{CP}^n$ without codimension-one multiple fibers yields holomorphic dominability by $\mathbb{C}^{2n}$. It further proves that abelian fibrations over log-Fano bases give dense entire curves, linking rational connectivity and Campana–Winkelmann theory to anti-hyperbolicity. The work relies on foundational hyperkähler geometry (BBF form, parabolic classes, SYZ conjectures), Néron-model techniques, and known results for Hilbert schemes and generalized Kummer varieties, providing concrete dominability results and paving the way for broader applications in hyperkähler and primitive symplectic geometry.

Abstract

By restricting to (a linear subspace of) an affine chart in projective space, a complex stably rational or unirational manifold of dimension $m$ is meromorphically dominable by $\mathbb C^m$, i.e., admits a meromorphic dominating map from $\mathbb C^m$. So are varieties that are birational to abelian varieties and Kummer K3 surfaces. G. Buzzard and the second author have shown that elliptic K3 surfaces are holomorphically dominable by $\mathbb C^2$, i.e. admitting a holomorphic map with nontrivial Jacobian. In this paper we explore various examples and criteria for meromorphic and holomorphic dominability by $\mathbb C^m$ of certain hyperkähler manifolds, generalizing some known results about K3 surfaces. Anti-hyperbolicity has several interpretations in the sense of vanishing of the Kobayashi-Royden metrics, admitting dense entire holomorphic curves, or dominating holomorphic or meromorphic maps from the complex affine space of the same dimension.