Rigidity of Elliptic Genera: From Number Theory to Geometry and Back. (16th August 2021)

• Record Type: Journal Article
• Authors: Bringmann, Kathrin; Caviedes Castro, Alexander; Sabatini, Silvia; Schwagenscheidt, Markus
• Is Part Of: International mathematics research notices. Volume 2022:Number 22(2022)
• Page Start: 17475
• Page End: 17533
• Publication Date: 2021-08-16
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab172
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: In this paper, we derive topological and number theoretical consequences of the rigidity of elliptic genera, which are special modular forms associated to compact almost complex manifolds. On the geometry side, we prove that rigidity implies relations between the Betti numbers and the index of a compact symplectic manifold 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 ${{\mathbb{C}}} P^n$ .