均质C21模型
Homogeneous C21 Models
摘要：
Fels-Kaup（Acta Mathematica 2008）对齐次$\mathfrak{C}_{2,1}$超曲面$M^5 \subset \mathbb{C}^3$进行了分类，并发现它们都在某个仿射齐次曲面$S^2 \subset \mathbb{R}^3$上双全纯于管$S^2 \times i \mathbb{R}^3$。第二和第三作者在2003.08166中，通过执行高度非直接的计算，进行了Cartan等效方法来分类与这种$\mathfrak{C}^{2,1}$超曲面$M^5 \subset \mathbb{C}^3$相关的PDE系统的齐次模型。Kolar-Kossovskiy 1905.05629和作者2003.01952构造了$\mathfrak{C}_{2,1}$超曲面$M^5 \subset \mathbb{C}^3$的形式和收敛的Poincaré-Moser规范型。但这只是第一步，初步的一步。事实上，由于计算上的障碍，Fels-Kaup分类的不变分支树在文献中仍然缺失。本文应用幂级数等价方法，验证了Fels-Kaup 2008的结论，并找到了一棵微分不变树。要终止中间（最粗）分支，需要使用$5$变量计算最高阶的$10$。同样，手工计算是不简单的。
英文摘要：
Fels-Kaup (Acta Mathematica 2008) classified homogeneous $\mathfrak{C}_{2,1}$ hypersurfaces $M^5 \subset \mathbb{C}^3$ and discovered that they are all biholomorphic to tubes $S^2 \times i \mathbb{R}^3$ over some affinely homogeneous surface $S^2 \subset \mathbb{R}^3$. The second and third authors in 2003.08166, by performing highly non-straightforward calculations, conducted the Cartan method of equivalence to classify homogeneous models of PDE systems related to such $\mathfrak{C}^{2,1}$ hypersurfaces $M^5 \subset \mathbb{C}^3$. Kolar-Kossovskiy 1905.05629 and the authors 2003.01952 constructed a formal and a convergent Poincaré-Moser normal form for $\mathfrak{C}_{2,1}$ hypersurfaces $M^5 \subset \mathbb{C}^3$. But this was only a first, preliminary step. Indeed, the invariant branching tree underlying Fels-Kaup's classification was still missing in the literature, due to computational obstacles. The present work applies the power series method of equivalence, confirms Fels-Kaup 2008, and finds a differential-invariant tree. To terminate the middle (thickest) branch, it is necessary to compute up to order $10$ with $5$ variables. Again, calculations, done by hand, are non-straightforward.
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