Classifying local orthogonal pairs and generalized Huang equations between hyperquadrics at the borderline dimension $n'=2(r+s)-2$

Gao (Math. Res. Lett. 2025) generalized Huang's Lemma — the central rigidity tool for proper holomorphic mappings between balls and hyperquadrics — by proving that solutions of $\sum_{j=1}^{n'}\varphi_j(z)\psi_j(z)=\|z\|^2_{r,s}H(z,\bar z)$ with $r+s\le n'\le 2(r+s)-3$ and $H\not\equiv 0$ have a rank-one multiplier $H=h_1h_2$ and partially linear structure, and correspondingly that local orthogonal pairs from $P^{r,s,t}$ into $P^{n',0}$ are quasi-standard or null. A generalized Whitney map shows that both conclusions fail at the borderline $n'=2(r+s)-2$, where the multiplier can have Hermitian rank greater than one, and no structural description is known there; likewise the classification of local orthogonal pairs between $P^{r,s}$ and indefinite targets $P^{r',s'}$ is available only in restricted ranges. The problem is to classify the local structure of orthogonal pairs (equivalently, of solutions of the generalized Huang equation) at this borderline codimension — whether every pair is null, quasi-standard, or of Whitney type — and to complete the classification for indefinite targets in the ranges not covered by existing rigidity theorems, which is exactly the input the transfer principle for Levi-degenerate sources consumes as a hypothesis.

Background

The rigidity of holomorphic mappings between bounded domains has been a central theme of several complex variables since Poincaré. A benchmark is proper holomorphic maps between unit balls $\mathbb{B}^n\to\mathbb{B}^N$ (maps whose preimage of every compact set is compact; between bounded domains this forces the boundary to be mapped to the boundary): Huang proved that a map which is $C^2$-smooth up to the boundary with $n\le N\le 2n-2$ is a totally geodesic (linear) embedding. The key tool, now called Huang's Lemma, states that if holomorphic functions $\varphi_j,\psi_j$ ($1\le j\le k\le n-1$) vanish at the origin and satisfy $\sum_{j=1}^{k}\varphi_j(z)\psi_j(z)=\|z\|^2 H(z,\bar z)$ with $H$ real-analytic, then $H\equiv 0$. The invariant controlling such identities is the Hermitian rank: a real-analytic function has Hermitian rank at most $p+q$ when it can be written as $\sum_{i=1}^{p}|f_i(z)|^2-\sum_{j=1}^{q}|g_j(z)|^2$ with holomorphic $f_i,g_j$, and the minimal such $p+q$ is its rank (a notion originating in Hilbert's 17th problem and its sums-of-squares analogues). Huang's Lemma says a multiplier of $\|z\|^2$ is forced to vanish in the low-term range; Xiao proved the first non-degenerate version: at term count $k=n$, either $H\equiv 0$ or $H$ has rank one and the tuples $\varphi,\psi$ are linear up to holomorphic factors.

The paper under consideration (Gao) works in a coordinate-free geometric framework. Let $\mathbb{C}^{r,s,t}$ denote $\mathbb{C}^{r+s+t}$ with the standard, possibly indefinite and degenerate Hermitian form $\langle z,w\rangle_{r,s,t}=\sum_{i\le r}z_i\bar w_i-\sum_{r<i\le r+s}z_i\bar w_i$ (the last $t$ coordinates enter trivially), and let $P^{r,s,t}$ be its projectivization. The generalized ball $B^{r,s,t}$ is its set of positive points; the boundary, consisting of null points, is a CR hypersurface (a real hypersurface of complex space whose tangent spaces carry a maximal complex subspace — the natural setting for boundary rigidity questions) which is Levi-degenerate when $t>0$ (the Levi form, the intrinsic pseudoconvexity curvature of such a hypersurface, is here the Hermitian form itself, which has a $t$-dimensional kernel along the boundary). A local orthogonal pair is a pair of holomorphic maps $(\varphi,\psi)$ from an open subset of $P^{r,s,t}$ to $P^{r',s',t'}$ such that $\varphi(p)\perp\psi(q)$ with respect to the target form whenever $p\perp q$ with respect to the source form; this is the projective incarnation of a pair of Segre maps (the Segre variety of a boundary point is the set of points joined to it by complex lines lying in the ambient hypersurface — a standard local invariant that linearizes CR mapping questions), and the diagonal case $\varphi=\psi=f$ is a local orthogonal map, the coordinate-free model of a local CR map between hyperquadrics. A pair is null if $\varphi(p)\perp\psi(q)$ for all $p,q$ (a trivial solution), standard if it is induced by a linear conformal pair of the underlying Hermitian spaces, and quasi-standard if it splits as a direct sum of a standard part and a null part. In homogeneous coordinates an orthogonal pair satisfies a generalized Huang equation $\sum_{j=1}^{r'}\varphi_j\psi_j-\sum_{j=r'+1}^{r'+s'}\varphi_j\psi_j=\|z\|^2_{r,s,t}H(z,\bar z)$, and the multiplier $H$ — determined up to a product of two holomorphic functions — completely characterizes the two rigid alternatives: the pair is null if and only if $H\equiv 0$, and quasi-standard if and only if $H=h_1(z)h_2(z)$ has Hermitian rank one.

Gao's main theorem extends Huang's Lemma to the sharp term range for arbitrary signature: if $r+s\le n'\le 2(r+s)-3$ and $H\not\equiv 0$, then $H=h_1h_2$ and the tuples are partially linear, $\varphi=B(h_1z_1,\dots,h_1z_{r+s},\Phi_1,\dots,\Phi_q)$ and $\psi=C(h_2z_1,\dots,h_2z_{r+s},\Psi_1,\dots,\Psi_q)$, with constant matrices satisfying $C^TB=H_{r,s,q}$ and $q=n'-r-s$. A companion transfer theorem deduces rigidity for Levi-degenerate sources from the non-degenerate case: if all local orthogonal pairs from $P^{r,s}$ to $P^{r',s'}$ are quasi-standard or null, then the same holds from $P^{r,s,t}$ for every $t,t'$. The numerical bound is sharp: at $n'=2(r+s)-2$ the generalized Whitney map of Huang–Xiao satisfies the equation with multiplier $(|z_1|^2+|z_n|^2)^2$ of Hermitian rank three (throughout, $n=r+s$ denotes the non-degenerate source dimension), so the rank-one conclusion, and with it quasi-standard rigidity, fails. The known rigidity patches are restricted: Gao's earlier theorem covers pairs from $P^{r,s}$ into positive-definite targets $P^{n',0}$ with $n'\le 2(r+s)-3$; Gao–Ng cover local orthogonal maps when $\min\{r',s'\}\le\min\{r,s\}$ (with $r,s\ge2$), when $\min\{r',s'\}\le 2\min\{r,s\}-2$ (with $t=0$), and when $r'+s'\le 2(r+s)-3$ (with $t=0$); Huang–Xiao independently proved the orthogonal-map transfer theorem for degenerate sources. No structural description exists at the borderline $n'=2(r+s)-2$ or for general indefinite targets outside these ranges — precisely the input that further applications of the transfer principle require as a hypothesis.

Problem Statement

Let $r,s,t\ge 0$ with $r+s\ge 2$, and set $n'=2(r+s)-2$. Classify the local orthogonal pairs (and, as the diagonal case, the local orthogonal maps) from $P^{r,s,t}$ into the positive-definite target $P^{n',0}$, up to the natural equivalence generated by linear conformal changes of source and target (that is, by linear conformal pairs in the sense of Definition 2.2 of the source paper: pairs of linear maps $\Phi,\Psi$ satisfying $\langle\Phi(z),\Psi(w)\rangle_{n',0}=\lambda\langle z,w\rangle_{r,s,t}$ for some $\lambda\in\mathbb{R}^*$) and by the direct-sum splitting into standard and null parts. Concretely, determine whether every such pair is null, quasi-standard, or equivalent to a generalized Whitney-type pair as in Example 1.2 of the source paper (the pair whose multiplier is the Hermitian-rank-three polynomial $(|z_1|^2+|z_n|^2)^2$ described in the background), or whether further local structures occur at this borderline codimension; in the non-degenerate case $t=0$ this amounts to classifying the solutions $(\{\varphi_j\}_{j=1}^{n'},\{\psi_j\}_{j=1}^{n'},H\not\equiv 0)$ of the generalized Huang equation $\sum_{j=1}^{n'}\varphi_j(z)\psi_j(z)=\|z\|^2_{r,s}H(z,\bar z)$ at term count $n'=2(r+s)-2$, including the possible Hermitian ranks of the multiplier, the uniqueness and moduli of the holomorphic factors, and the structure of the complementary components $\Phi_i,\Psi_i$. As the linked second component of the same objective — the missing hypothesis for the transfer theorem (Theorem 1.3 of the source) — determine for which signature pairs $(r',s')$ and which target dimensions beyond $2(r+s)-3$ every local orthogonal pair from the non-degenerate source $P^{r,s}$ to $P^{r',s'}$ is quasi-standard or null, and classify the exceptions when this dichotomy fails.

References

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