Structure of proper holomorphic maps between reducible bounded symmetric domains: rank-1 factors, general reducible targets, and semi-product maps beyond rank conditions

For proper holomorphic maps between reducible bounded symmetric domains, the known rigidity theory covers only restricted regimes: total geodesy when all irreducible factors of the domain have rank at least 2 and the ranks coincide (Chan 2020), a factorwise structure theorem when the target is a product of complex unit balls (Chan 2020, Theorem 3.11), and a classification of semi-product proper maps under the rank condition $r_{\min}(D)\ge r_{\max}(\Omega)$ (Seo 2018; Chan 2020, Theorem 4.5). The source paper itself exhibits nonstandard proper maps whenever the domain has a rank-1 (ball) factor, block-diagonal maps defeating the naive extension of the structure theorem to general reducible targets, and non-rational semi-product maps that are neither product maps nor totally geodesic. The 2019–2026 literature indicates that all three reducible-domain questions remain open: subsequent work addresses irreducible domains (Chan 2022; Kim–Mok–Seo 2025; Kim 2025; Gao 2025, 2026) or the isometric category (Kim–Seo 2024; Xiao 2022).

Background

A bounded symmetric domain is a bounded domain $D\subset\mathbb{C}^n$ every point of which is an isolated fixed point of a holomorphic involution of $D$; by the Harish-Chandra realization such domains are exactly the bounded circular domains underlying Hermitian symmetric spaces, and every bounded symmetric domain decomposes uniquely as a product $D=D_1\times\cdots\times D_k$ of irreducible bounded symmetric domains, its irreducible factors. The rank of $D$ is the maximal dimension of a polydisk embedded in $D$ as a product of one-dimensional symmetric subspaces; the domains of rank 1 are precisely the complex unit balls $B^n$, and the polydisk itself has all factors of rank 1. A proper holomorphic map is a holomorphic map under which preimages of compact sets are compact (for maps extending continuously to the boundary, this means the boundary is carried into the boundary). A totally geodesic holomorphic embedding is one whose image is a totally geodesic complex submanifold with respect to the canonical Kähler metrics (Bergman or suitably normalized Kähler–Einstein metrics) of domain and target; such a map is in particular an isometric embedding up to a positive normalizing constant. The guiding rigidity phenomenon is Tsai's theorem (1993): a proper holomorphic map from an irreducible bounded symmetric domain $D$ to a bounded symmetric domain $\Omega$ with $\operatorname{rank}(D)\ge\operatorname{rank}(\Omega)\ge2$ forces $\operatorname{rank}(D)=\operatorname{rank}(\Omega)$ and is a totally geodesic isometric embedding for the Bergman metrics up to a constant. For reducible domains one expects factorwise behavior: a product map is one of the form $f(Z^1,\dots,Z^k)=(f_1(Z^{\sigma(1)}),\dots,f_k(Z^{\sigma(k)}))$ for a permutation $\sigma$, each component depending on a single factor. Seo (2018) introduced the class of semi-product proper holomorphic maps: $f:D_1\times\cdots\times D_k\to\Omega_1\times\cdots\times\Omega_l$ is semi-product if for each domain factor $D_i$ there is a target factor $\Omega_j$ such that slicing the $j$-th component of $f$ in the $i$-th variable (with the other variables fixed at points of a dense open subset) yields a proper holomorphic map $D_i\to\Omega_j$; Seo proved that every rational proper map (all components rational functions) between reducible bounded symmetric domains is semi-product, and classified the equidimensional case, in which $\dim D=\dim\Omega$. Chan (2020) proved three complementary results: (i) Theorem 1.2 — a proper holomorphic map between bounded symmetric domains of equal rank $\ge2$, all of whose domain factors have rank $\ge2$, is a totally geodesic isometric embedding for the canonical Kähler metrics; (ii) Theorem 3.11 — if $D$ has rank at least 2, the target $\Omega=B^{m_1}\times\cdots\times B^{m_l}$ is a product of balls, and $f$ maps minimal disks of $D$ (one-dimensional rank-1 characteristic symmetric subspaces, i.e. the distinguished symmetric disk-orbits arising from the root decomposition of the symmetric space) properly into rank-1 characteristic symmetric subspaces of $\Omega$ (the symmetric subspaces of rank 1, each a ball), then $k\le l$ and $f$ decomposes factorwise as $(f_1(Z^1),\dots,f_k(Z^k),f_{k+1}(Z),\dots,f_l(Z))$ (with a product-map form when $k=l$), and moreover $D$ itself must be a product of balls; (iii) Theorem 4.5 — a semi-product proper map satisfying the rank condition $r_{\min}(D)\ge r_{\max}(\Omega)$ (where $r_{\min},r_{\max}$ are the smallest and largest ranks of the irreducible factors) satisfies $k\le l$, decomposes as above, is a product map when $k=l$, and is totally geodesic when additionally $r_{\max}(\Omega)\ge2$. The same paper shows why each hypothesis is load-bearing. Example 3.3 builds from D'Angelo's proper map $B^n\to B^{2n-1}$ a proper map $B^n\times D^I_{2,2}\to D^I_{3,2n+1}$ of rank 3 on both sides that is not an isometry for any canonical metrics, so Theorem 1.2 fails if even one domain factor is a ball. Example 3.12 exhibits a block-diagonal proper map $D^I_{p_1,q_1}\times\cdots\times D^I_{p_4,q_4}\to D^I_{p_1+p_2,q_1+q_2}\times D^I_{p_3+p_4,q_3+q_4}$ between domains of equal rank $\ge4$ (the rank of $D^I_{p,q}$ being $\min(p,q)$, with the index pairs chosen so that the two ranks agree, e.g. when within each block the minimum is attained on the same side) in which no component depends on a single factor, so the factorwise conclusion of Theorem 3.11 fails for targets that are not products of balls. Example 4.3 constructs semi-product maps $D^I_{2,2}\times D^I_{2,2}\to D^I_{3,3}\times D^I_{3,3}$, built from arbitrary bounded holomorphic functions and therefore generally non-rational, that are neither product maps nor totally geodesic even though $r_{\min}(D)=r_{\max}(\Omega)-1$, and Remark 4.6(2) gives non-totally-geodesic semi-product maps even under the rank condition when $k<l$.

Problem Statement

Let $D=D_1\times\cdots\times D_k$ and $\Omega=\Omega_1\times\cdots\times\Omega_l$ be bounded symmetric domains written as products of irreducible factors, and let $f:D\to\Omega$ be a proper holomorphic map. The solved regimes are: equal rank with all domain factors of rank $\ge2$ (total geodesy, Chan's Theorem 1.2); equidimensional pairs, i.e. $\dim D=\dim\Omega$ (Seo's classification); targets that are products of balls with minimal disks mapped properly into rank-1 characteristic symmetric subspaces (factorwise structure, Chan's Theorem 3.11); and semi-product maps under $r_{\min}(D)\ge r_{\max}(\Omega)$ (Chan's Theorem 4.5). Determine the structure of such maps in the three regimes left open by these results, which together form one objective — a rigidity and structure theory for proper holomorphic maps between reducible bounded symmetric domains:

(1) Domains with rank-1 factors. Since Example 3.3 gives proper maps of equal rank that are not totally geodesic once a domain factor is a ball, the naive extension of Theorem 1.2 is false. Identify sharp additional hypotheses on the pair $(D,\Omega)$, and on $f$, beyond $\operatorname{rank}(D)=\operatorname{rank}(\Omega)\ge2$, under which $f$ must be totally geodesic or must decompose factorwise as a product of proper maps between factors. Here sharp means necessary and sufficient: the hypotheses must be stated on $(D,\Omega)$ and $f$ without assuming the conclusion (so 'f is totally geodesic' or 'f is a product map' may not appear among them), must imply that conclusion, and must not be satisfied by the known non-rigid examples such as the map of Example 3.3.

(2) General reducible targets. Find the structure theorem that replaces Theorem 3.11 when $\Omega$ is reducible but not a product of complex unit balls — i.e., for bounded symmetric domains $D$ and $\Omega$ of the same rank $\ge2$, determine what decomposed or normal form is forced on a holomorphic map $f:D\to\Omega$ sending minimal disks of $D$ properly into rank-1 characteristic symmetric subspaces of $\Omega$ (here $f$, as in Theorem 3.11, is not assumed proper) — in a form that accounts for Example 3.12, in which the factorwise conclusion itself fails.

(3) Semi-product maps without rank dominance. Classify semi-product proper holomorphic maps $f:D\to\Omega$ without assuming $r_{\min}(D)\ge r_{\max}(\Omega)$, including the non-rational examples of Example 4.3; in particular determine which conclusions of Theorem 4.5 (the inequality $k\le l$, factorwise dependence of components, product-map form when $k=l$, and total geodesy when $r_{\max}(\Omega)\ge2$) survive, and under what corrected hypotheses.

The common scientific objective across regimes (1)–(3) is a complete description of when rigidity (total geodesy or product decomposition) holds for proper holomorphic maps between reducible bounded symmetric domains in these regimes.

References

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