Curve Classes on Calabi-Yau Complete Intersections in Toric Varieties

TL;DR

This paper proves the Integral Hodge Conjecture for curve classes on certain smooth varieties constructed as complete intersections in toric varieties, showing that their second homology is generated by rational curves.

Contribution

It establishes the Integral Hodge Conjecture for curve classes on smooth complete intersections in toric varieties, extending known cases and utilizing toric MMP techniques.

Findings

H_2(X,Z) is generated by rational curves

Integral Hodge Conjecture holds for these varieties

Applicable to smooth anticanonical hypersurfaces in toric Fano varieties

Abstract

We prove the Integral Hodge Conjecture for curve classes on smooth varieties of dimension at least three with nef anticanonical divisor constructed as a complete intersection of ample hypersurfaces in a smooth toric variety. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric MMP, we prove that in each case, $H_2(X,\mathbb{Z})$ is generated by classes of rational curves.