Compactification of complete Kähler manifolds of negative Ricci curvature (Q1113111)

The authors investigate the problem of compactification of complete Kähler manifolds of negative Ricci curvature. The main result is the following theorem: Let M be a complete Kähler manifold of dimension m and let \(\omega\) be the Kähler form. Assume the following: 1. \(Ric(\omega)<0\). 2. M is very strongly (m-2)-pseudoconcave. 3. The universal cover of M is Stein. Then M is biholomorphic to a quasiprojective manifold. Let us also mention that M is said to be very strongly (m-q)- pseudoconcave if there exists an infinite continuous exhaustion v whose complex Hessian \(\sqrt{-1}\partial {\bar \partial}\nu\) is seminegative and has at least q negative eigenvalues outside of a compact subset of M in the sense of distributions. The proof depends on the weak Riemann-Roch theorem for \(L^ 2\)-plurigeneral and the existence of Kähler-Einstein metrics. Also a purely differential geometric criterion for noncompact complete Kähler manifolds of finite volume to be quasiprojective is given.