Rigidity of elliptic genera: from number theory to geometry and back

Abstract

In this paper we derive topological and number theoretical consequences of the rigidity of elliptic genera, which are special modular forms associated to each compact almost complex manifold. In particular, on the geometry side, we prove that rigidity implies relations between the Betti numbers and the index of a compact symplectic manifold of dimension $2n$ admitting a Hamiltonian action of a circle with isolated fixed points. We investigate the case of maximal index and toric actions. On the number theoretical side we prove that from each compact almost complex manifold of index greater than one, that can be endowed with the action of a circle with isolated fixed points, one can derive non-trivial relations among Eisenstein series. We give explicit formulas coming from the standard action on $\mathbb{C} P^n$.