Compact Kihler manifolds with semi-positive Ricci curvature have been inves- tigated by various authors.From Peternell's work,if M is a compact K■hler n-manifold with semi-positive Ricci curvature and finite fu...Compact Kihler manifolds with semi-positive Ricci curvature have been inves- tigated by various authors.From Peternell's work,if M is a compact K■hler n-manifold with semi-positive Ricci curvature and finite fundamental group,then the universal cover has a decomposition ■≌X_1 x"'x X_m,where X_j is a Calabi-Yau manifold,or a hy- perKhler manifold,or X_j satisfies H^o(X_j,Ω~p)=O.The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Khler manifolds by us- ing the Gromov-Hausdorff convergence.Let M be a compact complex n-manifold with non-vanishing Euler number.If for any ■>O,there exists a K■hler structure(J_e,g_e)on M such that the volume Vol_(ge)(M)<V,the sectional curvature|K(g_e)|<A^2,and the Ricci-tensor Ric(g_e)>-■g_e,where V and A are two constants independent of ■.Then the fundamental group of M is finite,and M is diffeornorphic to a complex manifold X such that the universal covering of X has a decomposition, ■≌X_1x...xX_s,where X_i is a Calabi-Yau manifold,or a hyperK■ihler manifold,or X_i satisfies H^o(X_i,Ω~P)={O},p>O.展开更多
文摘Compact Kihler manifolds with semi-positive Ricci curvature have been inves- tigated by various authors.From Peternell's work,if M is a compact K■hler n-manifold with semi-positive Ricci curvature and finite fundamental group,then the universal cover has a decomposition ■≌X_1 x"'x X_m,where X_j is a Calabi-Yau manifold,or a hy- perKhler manifold,or X_j satisfies H^o(X_j,Ω~p)=O.The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Khler manifolds by us- ing the Gromov-Hausdorff convergence.Let M be a compact complex n-manifold with non-vanishing Euler number.If for any ■>O,there exists a K■hler structure(J_e,g_e)on M such that the volume Vol_(ge)(M)<V,the sectional curvature|K(g_e)|<A^2,and the Ricci-tensor Ric(g_e)>-■g_e,where V and A are two constants independent of ■.Then the fundamental group of M is finite,and M is diffeornorphic to a complex manifold X such that the universal covering of X has a decomposition, ■≌X_1x...xX_s,where X_i is a Calabi-Yau manifold,or a hyperK■ihler manifold,or X_i satisfies H^o(X_i,Ω~P)={O},p>O.
