Degree of Ball Maps with Maximum Geometric Rank

We will use a series of associated maps and state a series of normalization for a rational proper map. Write $\mathbb{H}_{n}=\{\,(z,w)\in\mathbb{C}^{n-1}\times\mathbb{C}\mid\operatorname{Im}w>\mathopen{}\mathclose{{}\left\lVert z}\right\rVert^{2}\,\}$ for the Siegel upper half-space and $\partial\mathbb{H}_{n}=\{\,(z,w)\in\mathbb{C}^{n-1}\times\mathbb{C}\mid\operatorname{Im}w=\mathopen{}\mathclose{{}\left\lVert z}\right\rVert^{2}\,\}$ for its boundary, the Heisenberg group. We parametrize $\partial\mathbb{H}_{n}$ by $(z,\bar{z},u)$ through the map $(z,\bar{z},u)\mapsto(z,u+i\mathopen{}\mathclose{{}\left\lVert z}\right\rVert^{2})$, and for a non-negative integer $m$ and a function $h(z,\bar{z},u)$ defined on a small ball $U$ around $0$ in $\partial\mathbb{H}_{n}$, we write $h=\operatorname{o_{wt}}(m)$ if $\frac{h(tz,t\bar{z},t^{2}u)}{\mathopen{}\mathclose{{}\left\lvert t}\right\rvert^{m}}\to 0$ uniformly for $(z,u)$ on every compact subset of $U$ as the real number $t\to 0$.

Let $F=(\tilde{f},\tilde{g})\colon\partial\mathbb{H}_{n}\to\partial\mathbb{H}_{N}$ be a rational CR map. Take any $p=(z_{0},w_{0})\in\partial\mathbb{H}_{n}$, consider $\sigma_{p}\in\operatorname{Aut}(\mathbb{H}_{n})$ and $\tau_{p}^{F}\in\operatorname{Aut}(\mathbb{H}_{N})$ given by

where $\cdot$ is the standard bilinear product. Then $F_{p}:=\tau_{p}^{F}\circ F\circ\sigma_{p}\colon\partial\mathbb{H}_{n}\to\partial\mathbb{H}_{N}$ is a rational CR map with $F_{p}(0)=0$. It follows from Huang’s pioneer paper [huang-1999-linearity]*Section 4 that there are automorphisms $H_{p},G_{p}\in\operatorname{Aut}(\mathbb{H}_{N})$ and $F_{p}^{*}:=H_{p}\circ F_{p}$ such that the rational CR map $F_{p}^{**}=(f_{p}^{**},\phi_{p}^{**},g_{p}^{**}):=G_{p}\circ F_{p}^{*}\colon\partial\mathbb{H}_{n}\to\partial\mathbb{H}_{N}$ satisfies the following normalization condition:

with $\overline{z}\cdot\overline{a_{p}^{\{1\}}(z)}\mathopen{}\mathclose{{}\left\lVert z}\right\rVert^{2}=\mathopen{}\mathclose{{}\left\lVert\phi_{p}^{\{2\}}(z)}\right\rVert^{2}$, where we denote by $h^{\{j\}}(z)$ a homogeneous polynomial degree $j$ in $z$, that is, $h^{\{j\}}(cz)=c^{j}h^{\{j\}}(z)$ for any $c\in\mathbb{C}$.

The normalization lets us write $a_{p}^{\{1\}}(z)=z\mathcal{A}_{p}$, where

is an $(n-1)\times(n-1)$ Hermitian positive semidefinite matrix. Write $\rho_{n}\colon\mathbb{H}_{n}\to\mathbb{B}_{n}$ for the Cayley transformation, which extends to $\rho_{n}\colon\partial\mathbb{H}_{n}\to\partial\mathbb{B}_{n}$.

Note that $0\leq\kappa_{0}\leq n-1$. Huang [huang-2003-semirigidity]*Lemma 2.2 (B) showed that $\kappa_{0}$ is independent of $H_{p}$ or $G_{p}$ and in fact depends only on the spherical equivalence class of $F$. The geometric rank is used to measure the degeneracy of the second fundamental form of the map. The minimum geometric rank $\kappa_{0}=0$ is associated with linear fractional maps [huang-1999-linearity].

To put the map $F$ into one more normal form, we write $\mathcal{S}_{0}=\{\,(j,k)\mid 1\leq j\leq\kappa_{0},1\leq k\leq n-1,j\leq k\,\}$, $\mathcal{S}_{1}=\{\,(j,k)\mid j=\kappa_{0}+1,k\in\{\,\kappa_{0}+1,\dots,N-n-P(n,\kappa_{0})\,\}\,\}$, $\mathcal{S}=\mathcal{S}_{0}\cup\mathcal{S}_{1}$, and $P(n,\kappa_{0})=\frac{\kappa_{0}(2n-\kappa_{0}-1)}{2}$, number of elements in $\mathcal{S}_{0}$. We can use Lemma 3.2 and its proof from Huang [huang-2003-semirigidity] to show that if $\operatorname{Rk}_{F}(0)=\kappa_{0}$, then $N\geq n+P(n,\kappa_{0})$ and there is an automorphism $\gamma_{p}\in\operatorname{Aut}(\mathbb{H}_{N})$ such that the rational CR map $F_{p}^{***}=(f,\phi,g):=\gamma_{p}\circ F_{p}^{**}\colon\partial\mathbb{H}_{n}\to\partial\mathbb{H}_{N}$ satisfies the following normalization condition:

Moreover, $\mu_{jk}=\sqrt{\mu_{j}+\mu_{k}}$ for $j,k\leq\kappa_{0},j\neq k$, and $\mu_{jk}=\sqrt{\mu_{j}}$ for $j\leq\kappa_{0}<k\text{ or }j=k\leq\kappa_{0}$.

Now let us focus on how to get a degree estimate from the normal forms.

Denote the Segre variety by $Q_{(\zeta,\eta)}=\{\,(z,w)\mid\frac{w-\bar{\eta}}{2i}=z\cdot\overline{\zeta}\,\}$. A useful result that we will use to find a degree estimate is due to Huang and Ji [huang-2001-mapping]*Lemma 5.4:

Writing $\beta_{p}:=\gamma_{p}\circ G_{p}\circ H_{p}\circ\tau_{p}^{F}$ gives us $F_{p}^{***}=\beta_{p}\circ F\circ\sigma_{p}$. Notice that

Since all automorphisms of $\operatorname{Aut}(\mathbb{H}_{N})$ are of degree $1$, so is $\beta_{p}$. We see that

which tells us that the condition $\deg F|_{Q_{p}}\leq K$ from Proposition 2.3 is equivalent to $\deg F_{p}^{***}|_{Q_{0}}\leq K$.

We summarize these results in the following. Write $\phi_{jk}^{(\ell)}$ and $\phi_{jk}^{(0)}$ for the coefficient of $z^{\ell}w$ and $w^{2}$ respectively in the Taylor series of $\phi_{jk}$. We get the following normal form up to 2nd order and degree result that will be useful to find degrees.

Let $F\colon\mathbb{B}_{n}\to\mathbb{B}_{N}$ be a proper holomorphic map that is $C^{3}$-smooth up to the boundary with geometric rank $\kappa_{0}$, $n\geq 2$, and $N=n+\frac{\kappa_{0}(2n-\kappa_{0}-1)}{2}$. Since $0\leq\kappa_{0}\leq n-1$, we get $N\leq\frac{n(n+1)}{2}$ and hence $F$ is a rational map by [huang-2005-several]*Corollary 1.4. Let $d$ be the degree of the map $F$. As the $\kappa_{0}<n-1$ case has been proved in [ji-2004-maps], we will assume $\kappa_{0}=n-1$.

The map $F$ is a rational proper map between balls and so extends holomorphically across the boundary $\partial\mathbb{B}_{n}$ due to a well-known result by Cima and Suffridge [cima-1990-boundary], and takes $\partial\mathbb{B}_{n}$ to $\partial\mathbb{B}_{N}$. Then the extension $\rho_{N}^{-1}\circ F\circ\rho_{n}\colon\partial\mathbb{H}_{n}\to\partial\mathbb{H}_{N}$, which we will also call $F$, is a rational CR map of degree $d$, as $\rho_{n}$ and $\rho_{N}^{-1}$ are rational maps of degree $1$. Take any $p\in\partial\mathbb{H}_{n}$ near the origin. Using Corollary 2.4 (ii) with $\kappa_{0}=n-1$, we get that $F$ is spherically equivalent to the origin preserving map

with

where $\lambda_{j}\neq 0$ for $j=1,\dots,n-1$, $\mu_{jk}>0$ for $(j,k)\in\mathcal{S}=\mathcal{S}_{0}$, and $\mathcal{S}_{1}=\varnothing$.

To prove that $d=\deg F\leq n+1$, it is sufficient to prove that for all $p\in\partial\mathbb{H}_{n}$, $\deg F^{**}_{p}|_{Q_{0}}\leq n+1$ because of Corollary 2.4 (iii). Here $Q_{0}=\{\,w=0\,\}$.

Consider the CR vector fields

for $k=1,\dots,n-1$ and complexify these to get

Write $\epsilon_{k}=2i\bar{\zeta}_{k}$ and compute

for $j,k=1,\dots,n-1$. On $\partial\mathbb{H}_{n}$, we get the basic equation $\operatorname{Im}g=\mathopen{}\mathclose{{}\left\lVert\tilde{f}}\right\rVert^{2}$, that is,

Complexification gives us

along any Segre variety $Q_{(\zeta,\eta)}$.

We will describe the normalized map $F_{p}^{***}$ along a Segre variety and get a degree estimate from there. Set $n^{\prime}=n-1$ and $N^{\prime}=N-n$. We apply $\mathcal{L}_{k}$ for $k\in[n^{\prime}]$ and $\mathcal{L}_{j}\mathcal{L}_{k}$ for $(j,k)\in\mathcal{S}_{0}$ on Equation 2 to get

Letting $(z,w)=0$ and $\eta=0$ gives us

that is, $\overline{g(\zeta,0)}=0$, and

Define

for $(j,k)\in\mathcal{S}_{0}$. We will label the components of $\phi$ both by single indices $\ell\in[N-n]$ and double indices $(j,k)\in\mathcal{S}_{0}$, and write $\iota(j,k)=\ell$ and $\iota^{-1}(\ell)=(j,k)$.

At $(0,0)$, we compute

for $k\in[n^{\prime}]$ and $(j,k)\in\mathcal{S}_{0}$.

Set $\phi^{(k)}=(\phi_{\iota^{-1}(1)}^{(k)},\dots,\phi_{\iota^{-1}(N^{\prime})}^{(k)})$ and $\phi^{(0)}=(\phi_{\iota^{-1}(1)}^{(0)},\dots,\phi_{\iota^{-1}(N^{\prime})}^{(0)})$, so that at $(0,0)$,

Notice that at $(0,0)$,

and write

We see that at $(0,0)$, $\det B=\prod_{k=1}^{n^{\prime}}\lambda_{\iota^{-1}(k)}=2^{n^{\prime}}\prod_{k=1}^{n^{\prime}}\mu_{\iota^{-1}(k)}>0$, so that near the origin, $\det B>0$ and $B$ is invertible. Hence

and

which describes $\tilde{f}$ along the Segre variety $Q_{0}$ in terms of the matrices $A$ and $B$. We see that $\det B$, $\operatorname{adj}B$, and $A$ are all polynomial maps in $\epsilon$ with

and

as polynomial maps in $\epsilon$, so in $\bar{\zeta}$.

We will in fact show in the next section that

1. (i)

   $\deg\det B\leq n$ and

2. (ii)

   $\deg(\operatorname{adj}B)A\leq n$,

so that $\deg\begin{bmatrix}(\det B)\epsilon\\ -(\operatorname{adj}B)A\epsilon\end{bmatrix}\leq n+1$, giving us $\deg\overline{\tilde{f}(\zeta,0)}\leq n+1$. Combining this with $\overline{g(\zeta,0)}=0$, we will have shown that $\deg\overline{F^{***}_{p}(\zeta,0)}\leq n+1$, that is, $\deg F^{***}_{p}|_{Q_{0}}\leq n+1$, proving our desired result.

We will look at the columns of linear and quadratic terms of matrix $B$ and those of matrix $A$. We define the matrix $E\in\mathbb{C}^{N^{\prime}\times n^{\prime}}$ and the column vector $e\in\mathbb{C}^{N^{\prime}}$ by defining $E_{jk}$ and $e_{j}$ in the following way: Write $(p,q)=\iota^{-1}(j)$, let $\delta$ be the Kronecker delta function, and define

We write $E_{k}$ for columns of $E$ to get that

Using 2, write $B=\sum_{k=0}^{2}B^{\{k\}}$. We see that the quadratic terms of $B$ form the matrix

so that all its columns are multiples of the column matrix $e$, making $B^{\{2\}}$ a determinant-$0$ rank-$1$ matrix. Moreover, as