  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.
