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

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.