Kobayashi Non-hyperbolicity of Calabi–Yau Manifolds Via Mirror Symmetry

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 Xˇ 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.