Kobayashi non-hyperbolicity of Calabi-Yau manifolds via mirror symmetry

TL;DR

This paper uses mirror symmetry to prove that all Calabi-Yau manifolds with a mirror dual that is not Hodge degenerate contain entire curves, showing they are Kobayashi non-hyperbolic.

Contribution

It establishes a link between mirror symmetry and Kobayashi non-hyperbolicity, demonstrating the existence of entire curves on Calabi-Yau manifolds via their mirror duals.

Findings

Existence of elliptic or rational curves on Calabi-Yau manifolds with non-Hodge degenerate mirrors

All such Calabi-Yau manifolds are Kobayashi non-hyperbolic

Higher-dimensional simply connected Calabi-Yau manifolds are not known to be Hodge degenerate

Abstract

A compact complex manifold is Kobayashi non-hyperbolic if there exists an entire curve on it. Using mirror symmetry we establish that there are (possibly singular) elliptic or rational curves on any Calabi-Yau manifold $X$, whose mirror dual $\check X$ exists and is not "Hodge degenerate", therefore proving that $X$ is Kobayashi non-hyperbolic. We are not aware of any higher dimensional simply connected Calabi-Yau manifolds that satisfy the "Hodge degenerate" condition.