Elliptic Genera, Torus Orbifolds and Multi-fans

A torus orbifold is an oriented closed orbifold of even dimension which admits an action of a torus of half the dimension of the orbifold with some orientation data concerning codimension two fixed point set components of circle subgroups and with some restrictions on isotropy groups of points of the orbifold. Typical examples are complete toric varieties with simplicial fan. To a toric variety there corresponds a fan, and that correspondence is ono-to-one. In particular algebro-geometric properties of a toric variety can be described in terms of combinatorial properties of the corresponding fan in principle, see e.g. [O]. In a similar way, to a torus orbifold there corresponds a multi-fan, a generalaization of the notion of fan. The notion of multi-fan was introduced in [M] and a combinatorial theory of multi-fans was developed in [HM]. In particular it was shown there that every complete simplicial multi-fan of dimension greater than two can be realized as the one associated with a torus orbifold. It should be noticed that a torus orbifold also determines a set of vectors which generates the one dimesnional cones of the associated multi-fan. It turns out that many topological invariants of a torus orbifold can be described in terms of the multi-fan and the set of generating vectors associated with it. The purpose of the present note is to discuss elliptic genera for torus orbifolds and multi-fans. Two sorts of elliptic genus are defined for stably almost complex orbifolds. One is the direct generalization of elliptic genus for stably almost complex manifolds to stably almost complex orbifolds which we shall denote by φ. The other, denoted by φ̂, is the so-called orbifold elliptic genus which has its origin in string theory. Correspondingly we can define two sorts of elliptic genus for a pair of complete simplicial multi-fan and generating vectors. Note that, though there may not be a stably almost complex complex structure on a torus orbifold in general, one can still define elliptic genera φ and φ̂ for pairs of complete simplicial multi-fan and generating vectors, and therefore define them for general torus orbifolds vice versa. Borisov and Libgober gave a beautiful formula for elliptic genus φ of complete nonsingular toric varieties in [BL1]. Theorem 3.3 and Theorem 3.4 describe similar formulae expressing the equivariant elliptic genera φ and φ̂ of a complete simplicial multi-fan as a virtual character of the associated torus. The starting point of [BL1] was the sheaf cohomology of toric varieties. Our starting point is the fixed point formula of the AtiyahSinger type due to Vergne applied to the action of the torus. A remarkable feature of elliptic genera is their rigidity property. If the circle group acts on a closed almost complex (or more generally stably almost complex) 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, that is, it is a constant character of the circle group. It was conjectured by Witten [W] and proved by Taubes [T], Bott-Taubes [BT] and Hirzebruch [H]. Liu [L] found a simple proof using the modularity of elliptic genera. Applying this to a non-singular complete toric variety we see that its elliptic genera φ