## Abstract

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})| < Λ^{2}, 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 ≃ X_{1} × ⋯ × 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.

Original language | English |
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Pages (from-to) | 421-428 |

Number of pages | 8 |

Journal | Chinese Annals of Mathematics. Series B |

Volume | 28 |

Issue number | 4 |

DOIs | |

Publication status | Published - Aug 2007 |

Externally published | Yes |

## Keywords

- Gromov-Hausdorff
- Kähler metric
- Ricci curvature