Rational curves on fibered Calabi-Yau manifolds

TL;DR

This paper proves that certain fibered Calabi-Yau manifolds with specific structures always contain rational curves, advancing understanding of their geometric properties and rational connectivity.

Contribution

It establishes the existence of rational curves on fibered Calabi-Yau manifolds with relatively trivial canonical bundle and elliptic fibrations, extending previous results.

Findings

Rational curves exist on fibered Calabi-Yau manifolds with finite fundamental group.

Calabi-Yau manifolds fibred over a curve with abelian variety fibers contain rational curves.

Results apply to manifolds with relatively trivial canonical bundle and elliptic fibrations.

Abstract

We show that a smooth projective complex manifold of dimension greater than two endowed with an elliptic fiber space structure and with finite fundamental group always contains a rational curve, provided its canonical bundle is relatively trivial. As an application of this result, we prove that any Calabi-Yau manifold that admits a fibration onto a curve whose general fibers are abelian varieties always contains a rational curve.