Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation
Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation
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卡勒变形下紧型不可约埃尔米特对称空间的刚性
DOI：
10.1007/s002220050209
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发表时间：
1996
影响因子：
3.1
通讯作者：
N. Mok
中科院分区：
文献类型：
--
作者：
Jun;N. Mok
我们研究了紧致型不可约厄米特对称空间S作为射影代数流形的变形，证明了复杂结构不可能发生跳跃。对于秩为2$的$S$，有一个相应的约化线性群$G$，使得$S$允许一个全纯的$G$-结构，对应于切丛的结构群的约化.$S是唯一的单连通紧致复流形，它允许这样一个同时可积的$G$-结构。为了证明$S$的变形刚性，充分证明了相应的可积$G$-结构收敛。
我们用有理曲线的形变理论进行矛盾论证。假设复杂结构发生跳跃，则特殊纤维$X_0$上与1次有理曲线相切的向量锥是线性退化的，从而定义了$X_0$上的一个适当的亚纯分布$W_$。我们证明了这样的$W$不可能存在。一方面，$W$的可积性与$b_2(X)=1$的事实相矛盾。另一方面，通过将$W$的积分复曲面族生成为1次有理曲线的铅笔，我们证明了$W$是自动可积的。为了验证是否有足够的积分曲面，我们需要对特殊纤维上的普通锥体进行描述。我们证明了它们实际上是标准圆锥在线性投影下的图像。我们通过研究在一点标记的极小有理曲线的Chow空间的正规化的变形来实现这一点，这些极小有理曲线本身是厄米特对称的，除了Grassmannians曲线是不可约的。
We study deformations of irreducible Hermitian symmetric spaces $S$ of the compact type, known to be locally rigid, as projective-algberaic manifolds and prove that no jump of complex structures can occur. For each $S$ of rank $\ge 2$ there is an associated reductive linear group $G$ such that $S$ admits a holomorphic $G$-structure, corresponding to a reduction of the structure group of the tangent bundle. $S$ is characterized as the unique simply-connected compact complex manifold admitting such a $G$-structure which is at the same time integrable. To prove the deformation rigidity of $S$ it suffices that the corresponding integrable $G$-structures converge.
We argue by contradiction using the deformation theory of rational curves. Assuming that a jump of complex structures occurs, cones of vectors tangent to degree-1 rational curves on the special fiber $X_0$ are linearly degenerate, thus defining a proper meromorphic distribution $W$ on $X_0$. We prove that such $W$ cannot possibly exist. On the one hand, integrability of $W$ would contradict the fact that $b_2(X)=1$. On the other hand, we prove that $W$ would be automatically integrable by producing families of integral complex surfaces of $W$ as pencils of degree-1 rational curves. For the verification that there are enough integral surfaces we need a description of generic cones on the special fiber. We show that they are in fact images of standard cones under linear projections. We achieve this by studying deformations of normalizations of Chow spaces of minimal rational curves marked at a point, which are themselves Hermitian symmetric, irreducible except in the case of Grassmannians.