Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities
Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities
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DOI：
10.2969/jmsj/06431005
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发表时间：
2009-06
影响因子：
0.7
通讯作者：
R. Goto
中科院分区：
文献类型：
--
作者：
R. Goto
设$X_0$是一个只有正规孤立奇点$p$的仿射簇，$\pi：X\to X_0$是奇点的光滑分解，其标准线丛为$K_X$。如果仿射簇X 0 p的补是Einstein-Sasakian流形S的锥C（S）=\Bbb R_{>0}\times S，则证明了X 0的可分解X在H2（X）的每个K\“ahler类中都存在完备的Ricci平坦K\“ahler度量.应用连续性方法求解Monge-Amp\'ere方程，得到了Ricci平坦圆锥K\' ahler度量的相关存在性定理和唯一性定理。利用Sasakian流形S上的上同调基本群的Hodge分解和Lefschetz分解，利用完备分解X上的消失定理，在每个K ahler类中构造了一个初始K ahler度量，并证明了Ricci平坦完备K ahler流形作为完备分解的存在性.
Let $X_0$ be an affine variety with only normal isolated singularity $p$ and $\pi: X\to X_0$ a smooth resolution of the singularity with trivial canonical line bundle $K_X$. If the complement of the affine variety $X_0\backslash\{p\}$ is the cone $C(S)=\Bbb R_{>0}\times S$ of an Einstein-Sasakian manifold $S$, we shall prove that the crepant resolution $X$ of $X_0$ admits a complete Ricci-flat K\"ahler metric in every K\"ahler class in $H^2(X)$. We apply the continuity method for solving the Monge-Amp\`ere equation to obtain a relevant existence theorem and a uniqueness theorem of Ricci-flat conical K\"ahler metrics. By using the vanishing theorem on the crepant resolution $X$ and the Hodge and Lefschetz decompositions of the basic cohomology groups on the Sasakian manifold $S$, we construct an initial K\"ahler metric in every K\"ahler class on which the existence theorem can be applied.We show there are many examples of Ricci-flat complete K\"ahler manifolds arising as crepant resolutions.