15 mar 2017 -- 15:00
Aula di Consiglio, Dip.Matematica, Università "La Sapienza", Roma
Abstract.
An affine manifold X (in the sense of differential geometry) is a differentiable manifold admitting an atlas of charts with value in an affine space with locally constant affine change of coordinates. Equivalently, it is a manifold admitting a flat torsion free connection on its tangent bundle. Around 1955 Chern asked if there is any topological obstruction to the existence of an affine structure on a compact manifold X. He conjectured that the Euler characteristic e(TX) of any compact affine manifold has to vanish. I will discuss this conjecture and a proof when X is special affine (i.e. X is affine and moreover admits a parallel volume form). Surprisingly (or not), the proof relies on algebraic methods coming from hypercomplex geometry.