Type-I 有界对称域的高秩间隙定理

中文摘要 AI 辅助

设 $\Omega_{r,s}$ 与 $\Omega_{r',s'}$ 为 Type-I 有界对称域，其中 $s>r$，$s'>r'$。设 $F:U\to M_{r',s'}(\mathbb C)$ 为全纯映射，满足 $\overline{\Omega}_{r,s}\subset U$，并假设 $F(S_{r,s})\subset S_{r',s'}$。若 \\[ k(s-r)\leq s'-r'<(k+1)(s-r) \quad\text{且}\quad r'>kr, \\] 则在目标坐标下，该映射包含一个大小为 $r'-kr$ 的固定恒等块。证明首先将边界方程转化为关于一阶导数的矩阵恒等式。直接维数计数给出 $g(s-r)\leq s'-r'$，其中 $g$ 是由该恒等式产生的半正定矩阵的秩。Shilov 边界的极值性质与标量 Hopf 边界引理进而表明映射的某些行是常数。这便给出了固定恒等块。我们还给出例子说明其大小是最优的，并解释为何证明需要 $F$ 定义在 $\overline{\Omega}_{r,s}$ 的邻域上。

英文摘要

Let $Ω_{r,s}$ and $Ω_{r',s'}$ be Type-I bounded symmetric domains with $s>r$ and $s'>r'$. Let $F:U\to M_{r',s'}(\mathbb C)$ be holomorphic, where $\overlineΩ_{r,s}\subset U$, and suppose that $F(S_{r,s})\subset S_{r',s'}$. If \[ k(s-r)\leq s'-r'<(k+1)(s-r) \quad\text{and}\quad r'>kr, \] then, up to target coordinates, the map contains a fixed identity block of size $r'-kr$. The proof first turns the boundary equation into a matrix identity for the first derivative. A direct dimension count gives $g(s-r)\leq s'-r'$, where $g$ is the rank of a positive semidefinite matrix arising from this identity. The extremal property of the Shilov boundary and the scalar Hopf boundary lemma then show that certain rows of the map are constant. This gives the fixed identity block. We also give examples showing that its size is optimal and explain why the proof needs $F$ on a neighborhood of $\overlineΩ_{r,s}$.

发表机构

• School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）

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