Projective structures on moduli spaces of compact complex hypersurfaces

Abstract

It is shown that moduli spaces of complete families of compact complex hypersurfaces in complex manifolds often come equipped canonically with projective structures satisfying some natural integrability conditions. 1. PROJECTIVE CONNECTIONS Let M be a complex manifold. Consider the following equivalence relation on the set of affine torsion-free connections on M: two connections F and F are said to be projectively equivalent if they have the same geodesics, considered as unparameterized paths. In a local coordinate chart fte}, a = 1,..., dimM on MI where F and F are represented by Christoffel symbols 1F' and 1F' respectively, this equivalence relation reads [H]