An analog of Chern’s conjecture for the Euler–Satake characteristic of affine orbifolds

Abstract

S.S. Chern conjectured that the Euler characteristic of every closed affine manifold has to vanish. We present an analog of this conjecture stating that the Euler–Satake characteristic of any compact affine orbifold is equal to zero. We prove that Chern’s conjecture is equivalent to its analog for the Euler–Satake characteristic of compact affine orbifolds, not necessarily effective. This fact allowed us to extend to orbifolds sufficient conditions for Chern’s conjecture proved by Klingler and Kostant–Sullivan. Thus, we prove that, if an n-dimensional compact affine orbifold N is complete or if its holonomy group belongs to the special linear group SL(n,R), then the Euler–Satake characteristic of N has to vanish. An application to pseudo-Riemannian orbifolds is considered. We give examples of orbifolds from the class under investigation. In particular, we construct an example of a compact incomplete affine orbifold with vanishing Euler–Satake characteristic, the holonomy group of which is not contained in SL(n,R).