Elliptic genera, torus manifolds and multi-fans

The rigidity theorem of Witten-Bott-Taubes-Hirzebruch tells us that, if the circle group acts on a closed almost complex (or more generally unitary) manifold whose first Chern class is divisible by a positive integer N greater than 1, then its equivariant elliptic genus of level N is rigid. Applying this to a non-singular compact toric variety, we see that its elliptic genus of level N is rigid if its first Chern class is divisible by N. But, using a vanishing theorem of Hirzebruch, we can show moreover that the genus actually vanishes. In this note we shall extend this result to torus manifolds. In fact, a non-singular complete multi-fan is associated with a torus manifold, and rigidity and vanishing of elliptic genus of level N can be formulated and proved for non-singular complete multi-fans. Some applications on compact non-singular toric varieties with the first Chern class divisible by a large positive integer are given.