Kähler Manifolds with Almost Non-negative Ricci Curvature

Abstract

Compact Kahler manifolds with semi-positive Ricci curvature have been investigated by various authors. From Peternell’s work, if M is a compact Kahler n-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{M} \cong X_{1} \times \cdots \times X_{m} \), where Xj is a Calabi-Yau manifold, or a hyperKahler manifold, or Xj satisfies H0(Xj , Ωp) = 0. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Kahler 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 Kahler structure (J∈, g∈) on M such that the volume \( {\text{Vol}}_{{g_{ \in } }} {\left( M \right)} −∈g∈, 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, \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{X} \cong X_{1} \times \cdots \times X_{s} \), where Xi is a Calabi-Yau manifold, or a hyperKahler manifold, or Xi satisfies H0(Xi , Ωp) = {0}, p > 0.