On sharp rates and analytic compactifications of asymptotically conical Kähler metrics
On sharp rates and analytic compactifications of asymptotically conical Kähler metrics
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关于渐近圆锥凯勒度量的锐率和解析紧化
DOI：
10.1215/00127094-2019-0073
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发表时间：
2014
影响因子：
2.5
通讯作者：
Chi Li
中科院分区：
文献类型：
--
作者：
Chi Li
设$X$是复流形，$S右行X$是复子流形的嵌入。假设嵌入是$(k-1)$-可线性化或$(k-1)$-舒适嵌入的，通过对法锥的变形，我们构造了从法丛$N_{S}$中零截面的一个小邻域到$X$中的$S$的一个小邻域的微分同胚映射$F$，使得$F$在精确意义下全纯到$(k-1)$阶.利用这个$F，我们得到了Tian-Yau构造的渐近锥Calabi-Yau度量的渐近速度的最优估计。此外，当$S$是满足适当上同调条件的充分因子时，我们将舒适嵌入的阶与由变形产生的法向孤立锥奇点到法锥的变形的权重联系起来。我们还给出了一个例子，说明舒适嵌入的条件取决于分离式提升。然后，我们证明了复锥上复杂结构的变形的一个解析紧致结果，该复锥在无穷远处衰减到任何正阶。这可以看作是Pinkham关于具有负权的锥奇点变形的结果的解析对应。
Let $X$ be a complex manifold and $S\hookrightarrow X$ be an embedding of complex submanifold. Assuming that the embedding is $(k-1)$-linearizable or $(k-1)$-comfortably embedded, we construct via the deformation to the normal cone a diffeomorphism $F$ from a small neighborhood of the zero section in the normal bundle $N_{S}$ to a small neighborhood of $S$ in $X$ such that $F$ is in a precise sense holomorphic to the $(k-1)$-th order. Using this $F$ we obtain optimal estimates on asymptotical rates for asymptotically conical Calabi-Yau metrics constructed by Tian-Yau. Furthermore, when $S$ is an ample divisor satisfying an appropriate cohomological condition, we relate the order of comfortable embedding to the weight of the deformation of the normal isolated cone singularity arising from the deformation to the normal cone. We also give an example showing that the condition of comfortable embedding depends on the splitting liftings. We then prove an analytic compactification result for the deformation of the complex structure on a complex cone that decays to any positive order at infinity. This can be seen as an analytic counterpart of Pinkham's result on deformations of cone singularities with negative weights.