论文标题
在对称Bidisc的几何形状上
On the geometry of the symmetrized bidisc
论文作者
论文摘要
我们研究了$ 2 $复杂的尺寸歧管歧管bidisc $ \ mathbb {g} $在本身上的$ 2 $复数歧管歧管的行动。自动形态组为3个实际维度。它将$ \ mathbb {g} $放入所有的叶子中，所有这些都是3个真实的超曲面，除了一个，即皇家品种。这使我们研究了Isaev对所有Kobayashi-Hyperbolic 2复合尺寸歧管的分类，为此，全体形态自动形态的群体具有Isaev研究的实际维度3。确实，我们在对称的bidisc和域之间产生生物形态 \ [\ {（Z_1，Z_2）\ in \ Mathbb {C} ^2：1+ | Z__1 | ^2- | Z_____2 | ^2> | 1+ Z_1 ^2 -z_2 ^2 -Z_2 ^2 |，IM（Z_1（Z_1（1+ \ edline {Z_2}}}}）） 在Isaev的列表中。 Isaev称其为$ \ Mathcal D_1 $。 Biholomorthism的道路上有有关$ \ Mathbb {G} $的各种几何见解。 生物形态的几个后果随之而来，其中包括对对称Bidisc的两个新特征以及$ \ Mathcal D_1 $的几种新特征。在$ \ Mathcal D_1 $的结果中，特别感兴趣的是$ \ Mathcal D_1 $是“对称”。当我们对称（在上一节中的上下文中适当定义）$ω_1$或$ \ nathcal {d}^{（2）} _ 1 $（Isaev的符号）时，我们得到$ \ Mathcal d_1 $。这两个域$ω_1$和$ \ MATHCAL {d}^{（2）} _ 1 $在Isaev的列表中，他提到这些是Biholomormormormorphic to $ \ MathBb {D} \ Times \ Times \ Mathbb {D} $。我们在这些域之间产生明确的生物形态和$ \ mathbb {d} \ times \ mathbb {d} $。
We study the action of the automorphism group of the $2$ complex dimensional manifold symmetrized bidisc $\mathbb{G}$ on itself. The automorphism group is 3 real dimensional. It foliates $\mathbb{G}$ into leaves all of which are 3 real dimensional hypersurfaces except one, viz., the royal variety. This leads us to investigate Isaev's classification of all Kobayashi-hyperbolic 2 complex dimensional manifolds for which the group of holomorphic automorphisms has real dimension 3 studied by Isaev. Indeed, we produce a biholomorphism between the symmetrized bidisc and the domain \[\{(z_1,z_2)\in \mathbb{C} ^2 : 1+|z_1|^2-|z_2|^2>|1+ z_1 ^2 -z_2 ^2|, Im(z_1 (1+\overline{z_2}))>0\}\] in Isaev's list. Isaev calls it $\mathcal D_1$. The road to the biholomorphism is paved with various geometric insights about $\mathbb{G}$. Several consequences of the biholomorphism follow including two new characterizations of the symmetrized bidisc and several new characterizations of $\mathcal D_1$. Among the results on $\mathcal D_1$, of particular interest is the fact that $\mathcal D_1$ is a "symmetrization". When we symmetrize (appropriately defined in the context in the last section) either $Ω_1$ or $\mathcal{D}^{(2)}_1$ (Isaev's notation), we get $\mathcal D_1$. These two domains $Ω_1$ and $\mathcal{D}^{(2)}_1$ are in Isaev's list and he mentioned that these are biholomorphic to $\mathbb{D} \times \mathbb{D}$. We produce explicit biholomorphisms between these domains and $\mathbb{D} \times \mathbb{D}$.
