Curve classes on Calabi-Yau complete intersections in toric varieties

We prove the integral Hodge conjecture for curve classes on smooth varieties of dimension at least three constructed as a complete intersection of ample hypersurfaces in a smooth projective toric variety, such that the anticanonical divisor is the restriction of a nef divisor. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric minimal model program, we prove that in each case, H2(X,Z)$H_2(X,\mathbb {Z})$ is generated by classes of rational curves.