Kähler manifolds with almost non-negative Ricci curvature

Compact Kähler manifolds with semi-positive Ricci curvature have been investigated by various authors. From Peternell's work, if M is a compact Kähler n-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition M̃ ≃ X ₁ × ⋯ × X_m, where X _j is a Calabi-Yau manifold, or a hyperKähler manifold, or X _j satisfies H ⁰(X _j , Ω ^p ) = 0. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Kähler manifolds by using the Gromov-Hausdorff convergence. Let M be a compact complex n-manifold with non-vanishing Euler number. If for any > 0, there exists a Kähler structure (J_ε, g _ε) on M such that the volume Vol_{g ∈} (M) < V, the sectional curvature |K(g_ε)| < Λ², and the Ricci-tensor Ric(g_ε)> - g_ε, where V and Λ are two constants independent of . Then the fundamental group of M is finite, and M is diffeomorphic to a complex manifold X such that the universal covering of X has a decomposition, X̃ ≃ X₁ × ⋯ × X_s, where X _i is a Calabi-Yau manifold, or a hyperKähler manifold, or X _i satisfies H ⁰(X _i , Ω ^p ) = {0}, p > 0.