Orbifold elliptic genera and rigidity

Three types of rigidity theorem for orbifold elliptic genus of level N are proved. The first type deals with the case where N is relatively prime to the orders of all isotropy groups. If the top exterior power of the tangent bundle is divisible by N in the Picard group of orbifold line bundles, then the ofbifold genus of level N suitably modified has rigidity property with respect to compact connected group actions. The second type deals with the divisibility within the Picard group of genuine line bundles. In this case the orbifold elliptic genus itself has rigidity property.