On a conjecture of Oguiso about rational curves on Calabi-Yau threefolds

TL;DR

This paper proves that Calabi-Yau threefolds with a specific nef divisor and second Betti number greater than 4 necessarily contain rational curves, advancing understanding of their geometric structure.

Contribution

It establishes a new criterion linking nef divisors and the existence of rational curves on Calabi-Yau threefolds with high second Betti number.

Findings

Calabi-Yau threefolds with certain nef divisors contain rational curves

The second Betti number condition is crucial for the result

Provides insight into the geometry of Calabi-Yau threefolds

Abstract

Let X be a Calabi-Yau threefold. We show that if there exists on X a non-zero nef non-ample divisor then X contains a rational curve, provided its second Betti number is greater than 4.