Elliptic Genera, Torus Manifolds and Multi-fans

The rigidity theorem of Witten-Bott-Taubes-Hirzebruch [W, BT, H] 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 [H], we can show moreover that the genus actually vanishes. In this note we shall extend this result to torus manifolds [M, HM]. A torus manifold is an oriented closed manifold of even dimension which admits an action of a torus of half the dimenstion of the manifold with some orientation data concerning codimension two fixed point set components of circle subgroups. Though there is not an almost complex nor unitary structure on a torus manifold in general, one can still define elliptic genus of level N on torus manifolds. In the almost complex or unitary case the elliptic genus is defined as the index of a twisted Dirac operator, so that the equivariant index is a virtual character of the torus acting on the manifold. For general torus manifolds the equivariant elliptic genus of level N is defined in terms of the multifan associated with the torus manifold. Thus the genus is in fact defined for multi-fans (precisely speaking for complete simplicial multi-fans). The fact that it is a character of a torus is proved by using the multiplicity formula for Duistermaat-Heckman function for multi-polytopes given in [HM]. The Chern class is also defined for multi-fans, and rigidity and vanishing of elliptic genus of level N can be formulated and proved. One of the main results is Corollary 4.4 which states that, if the first Chern class of a complete non-singular multi-fan is divisible by N , then the elliptic genus of level N vanishes. The proof of rigidity and vanishing of level N elliptic genus follows the idea of the proof given in [H]. The formula for Duistermaat-Heckman function is also used to modify topological terms into combinatorial terms so as to be applicable to multi-fans. When N = 2, the torus manifold is a spin manifold. The corresponding multi-fan might be called a spin multi-fan. As a corollary we see that its signature vanishes in this case. The Ty-genus can be considered as a special value of equivariant elliptic genus. Thus, if the first Chern class is divisible by N , then Ty-genus vanishes for (−y) N = 1. One can derive some applications from this fact. For example if ∆ is a complete non-singular multifan of dimension n with first Chern class c1(∆) divisible by N and with non-vanishing Todd genus, then N must be equal or less than n+ 1 (Proposition 5.2). In the extremal case N = n + 1, if ∆ is assumed to be a complete non-singular ordinary fan, then ∆ must be isomorphic to the fan of projective space P. Hence a complete non-singular toric variety M of dimension n with c1(M) divisible by n + 1 must be isomorphic to P n as toric varity (Corollary 5.4). We show furthermore that, in case c1(M) is divisible by n, M is isomorphic to a certain projective space bundle over P (Corollary 5.8).