关于球管超曲面的仿射等价类数
On the Number of Affine Equivalence Classes of Spherical Tube Hypersurfaces
摘要：
我们考虑$\CC^{n+1}$中的Levi非退化管超曲面是$(k,n-k)$ -球面的，即局部CR-等价于具有签名$(k,n-k)$的Levi形式的超二次曲面，具有$n\le 2k$ .我们证明了在下列情况下，这类超曲面的仿射等价类的个数是无穷的（事实上是不可数的）：（i）$k=n-2$，$n\ge 7$ ; \linebreak;（ii）$k=n-3$，$n\ge 7$ ;（iii）$k\le n-4$ .对于$k$和$n$的所有其他值，除了$k=3$，$n=6$，仿射类的数量已知是有限的。例外情况$k=3$，$n=6$最近已经解决了费尔斯和Kaup谁给了一个家庭的$(3,3)$ -球形管超曲面的例子，其中包含不可数许多成对仿射非等价元素。本文用不同的方法处理Fels-Kaup的例子。本文给出了Fels-Kaup族超曲面的球性的直接证明，并利用$j$ -不变量证明了该超曲面族确实包含不可数的两两仿射非等价超曲面子族.
英文摘要：
We consider Levi non-degenerate tube hypersurfaces in $\CC^{n+1}$ that are $(k,n-k)$-spherical, i.e. locally CR-equivalent to the hyperquadric with Levi form of signature $(k,n-k)$, with $n\le 2k$. We show that the number of affine equivalence classes of such hypersurfaces is infinite (in fact, uncountable) in the following cases: (i) $k=n-2$, $n\ge 7$;\linebreak (ii) $k=n-3$, $n\ge 7$; (iii) $k\le n-4$. For all other values of $k$ and $n$, except for $k=3$, $n=6$, the number of affine classes is known to be finite. The exceptional case $k=3$, $n=6$ has been recently resolved by Fels and Kaup who gave an example of a family of $(3,3)$-spherical tube hypersurfaces that contains uncountably many pairwise affinely non-equivalent elements. In this paper we deal with the Fels-Kaup example by different methods. We give a direct proof of the sphericity of the hypersurfaces in the Fels-Kaup family, and use the $j$-invariant to show that this family indeed contains an uncountable subfamily of pairwise affinely non-equivalent hypersurfaces.
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