On anti-hyperbolicity for hyperk\"ahler varieties

TL;DR

This paper investigates the conditions under which hyperk"ahler manifolds, including K3 surfaces and related varieties, are dominable by complex affine spaces, highlighting their anti-hyperbolic properties and generalizing known results.

Contribution

It provides new criteria and examples for meromorphic and holomorphic dominability of hyperk"ahler manifolds, extending previous work on K3 surfaces.

Findings

Hyperk"ahler manifolds can be meromorphically dominated by ^m under certain conditions.

Examples include varieties birational to abelian varieties and Kummer K3 surfaces.

Anti-hyperbolicity relates to vanishing Kobayashi-Royden metrics and dense entire curves.

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 hyperk\"ahler 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.