Curvature properties and Shafarevich conjecture for toroidal compactifications of ball quotients

We study toroidal compactifications of finite volume complex hyperbolic manifolds. We obtain results on the existence or nonexistence of K\"ahler metrics satisfying certain nonpositive curvature properties on these compactifications. Starting from quotients of complex hyperbolic space by deep enough non-uniform arithmetic lattices, we also verify the Shafarevich conjecture for their compactifications, by showing that their universal covers are Stein.Comment: 26 pages, comments welcome

Propriétés de courbure et conjecture de Shafarevich pour les compactifications toroidales de quotients de la boule

We study toroidal compactifications of finite volume complex hyperbolic manifolds. We obtain results on the existence or nonexistence of K\"ahler metrics satisfying certain nonpositive curvature properties on these compactifications. Starting from quotients of complex hyperbolic space by deep enough non-uniform arithmetic lattices, we also verify the Shafarevich conjecture for their compactifications, by showing that their universal covers are Stein