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# Asian Journal of Mathematics

## Volume 22 (2018)

### Number 2

### Special issue in honor of Ngaiming Mok (1 of 3)

Guest Editors: Jun-Muk Hwang, Korea Institute for Advanced Study; Yum-Tong Siu, Harvard University; Wing-Keung To, National University of Singapore; Stephen S.-T. Yau, Tsinghua University; Sai-Kee Yeung, Purdue University

### On rational proper mappings among generalized complex balls

Pages: 355 – 380

DOI: https://dx.doi.org/10.4310/AJM.2018.v22.n2.a11

#### Authors

#### Abstract

We introduce the notion of *multiplier*, a real-valued bihomogeneous polynomial $M_F \in \mathbb{C}[z_1, \overline{z}_1, \dotsc , z_{r+s} , \overline{z}_{r+s}]$ canonically associated to a rational proper map $F$ from a generalized ball $D_{r,s}$ to another generalized ball. We prove that the multiplier $M_F$ 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 $D_{2,2}$ to an arbitrary $D_{r,s}$, demonstrating first of all that one may confine itself to the cases where ${r, s} \geq 2$ and $r + s \leq 10$ without loss of generality. Then, we show that for each maximal case, i.e. whenever $r + s = 10$, there exists a real one-parameter family of non-equivalent degree-$2$ holomorphic proper maps. Finally, we give a complete description of all degree-$2$ rational proper maps from $D_{2,2}$ to $D_{3,3}$, which is the minimal case where there are non-standard mappings.

#### Keywords

proper mappings, generalized balls, bounded symmetric domains, flag domains

#### 2010 Mathematics Subject Classification

32H35, 32M15, 32V15

Received 1 December 2016

Accepted 8 June 2017

Published 15 June 2018