On Abelian Families and Holomorphic Normal Projective Connections

1.) Pm(C), 2.) smooth abelian families, 3.) manifolds with universal covering Bm(C). Here Bm(C) denotes the ball in C , the non compact dual of Pm(C) in the sense of hermitian symmetric spaces. The second point inlcudes the flat case of an abelian manifold. Any compact Riemann surface admits a holomorphic normal projective connection, this is the famous uniformization theorem. Kobayashi and Ochiai showed that the list of projective surfaces with a holomorphic normal projective connection consists of P2(C), abelian surfaces and ball quotients ([KO80]). The above list was confirmed in the case of projective threefolds in [JR04]. Of particular interest are the manifolds with a holomorphic normal projective connection of intermediate Kodaira dimension 0 < κ(M) < m. The type we expect are locally symmetric spaces obtained as quotients of