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| K?hler Manifolds with Almost Non-negative Ricci Curvature |
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Citation： |
Yuguang ZHANG.K?hler Manifolds with Almost Non-negative Ricci Curvature[J].Chinese Annals of Mathematics B,2007,28(4):421~428 |
| Page view： 1161
Net amount： 1121 |
Authors： |
Yuguang ZHANG; |
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| Abstract： |
Compact K\"{a}hler manifolds with semi-positive Ricci curvature have been investigated by various authors. From Peternell's work, if $M$ is a compact K\"{a}hler $n$-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition $\widetilde{M}\cong X_{1}\times \cdots \times X_{m}$, where $X_{j}$ is a Calabi-Yau manifold, or a hyperK\"{a}hler manifold, or $X_{j}$ satisfies $H^{0}(X_{j}, \Omega^{p})=0$. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature K\"{a}hler manifolds by using the Gromov-Hausdorff convergence. Let $ M$ be a compact complex $n$-manifold with non-vanishing Euler number. If for any $\epsilon>0$, there exists a K\"{a}hler structure $(J_{\epsilon}, g_{\epsilon})$ on $M$ such that the volume ${\rm Vol}_{g_{\epsilon}}(M) < V$, the sectional curvature $|K (g_{\epsilon})|< \Lambda^{2}$, and the Ricci-tensor {\rm Ric}($g_{\epsilon}$)$> - \epsilon g_{\epsilon}$, where $V$ and $\Lambda$ are two constants independent of $\epsilon$. 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, $\wt{X }\cong X_{1} \times \cdots \times X_{s}$, where $X_{i}$ is a Calabi-Yau manifold, or a hyperK$\ddot{\rm a}$hler manifold, or $X_{i}$ satisfies $ H^{0} (X_{i}, \Omega^{p})=\{0\}$, $p>0$. |
Keywords： |
Gromov-Hausdorff, Ricci curvature, K?hler metric |
Classification： |
53C55, 53C21 |
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