On rational proper mappings among generalized complex balls
On rational proper mappings among generalized complex balls
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DOI：
10.4310/ajm.2018.v22.n2.a11
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发表时间：
2018
影响因子：
0.6
通讯作者：
Yung Gao;Sui-Chung Ng
中科院分区：
文献类型：
--
作者：
Yung Gao;Sui-Chung Ng
本文引入了乘子的概念，即实值双齐次多项式MF ∈ C[z1，z <$1，. . .，zr+s，z <$r+s]与从广义球Dr，s到另一广义球的有理真映射F正则相关.证明了乘子MF本质上决定映射F，从而可以通过乘子研究广义球间有理真映射的结构。本文利用乘子研究了从D2，2到任意Dr，s的二次有理真映射，首先证明了我们可以把自己限制在r，s ≥ 2和r + s ≤ 10的情况下，而不失一般性.然后，我们证明了，对于每个极大的情况下，即当r + s = 10时，存在一个实参数族的非等价的2次全纯真映射。最后，我们给出了从D2，2到D3，3的所有二次有理真映射的完整刻画，这是存在非标准映射的最小情形。
We introduce the notion of multiplier, a real-valued bihomogeneous polynomial MF ∈ C[z1, z̄1, . . . , zr+s, z̄r+s] canonically associated to a rational proper map F from a generalized ball Dr,s to another generalized ball. We prove that the multiplier MF essentially determines the map F and hence one can study the structure of rational proper mappings among generalized balls through the multiplier. We use the multiplier to study degree-2 rational proper maps from D2,2 to an arbitrary Dr,s, demonstrating first of all that one may confine itself to the cases where r, s ≥ 2 and r + s ≤ 10 without loss of generality. Then, we show that for each maximal case, i.e. whenever r + s = 10, there exists a real-parameter family of non-equivalent degree-2 holomorphic proper maps. Finally, we give a complete description of all degree-2 rational proper maps from D2,2 to D3,3, which is the minimal case where there are non-standard mappings.