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Proper holomorphic maps between bounded symmetric domains with small rank differences

Sung-Yeon Kim, Ngaiming Mok, Aeryeong Seo

TL;DR

This work proves rigidity for proper holomorphic maps between irreducible bounded symmetric domains with small rank differences. By combining boundary analysis, moduli spaces of subgrassmannians, VMRT/CR geometry, and rigidity results for admissible pairs of subvarieties, it derives a canonical factorization: $f = ι \circ F$ with $F = F_1 \times F_2$ where $F_1$ is a standard embedding and $ι$ a totally geodesic isometric embedding, and further identifies cases with nonexistence. The approach hinges on induced moduli maps, their rational extensions, and a sequence of rigidity steps (subgrassmannian respect, induced moduli map rigidity) that culminate in a global diagonalization of $f$. These results connect geometric structure theory with boundary behavior of holomorphic maps, offering a robust framework for rigidity in the setting of Hermitian symmetric domains and their compact duals.

Abstract

In this paper we study the rigidity of proper holomorphic maps $f\colon Ω\toΩ'$ between irreducible bounded symmetric domains $Ω$ and $Ω'$ with small rank differences: $2\leq \text{rank}(Ω')< 2\,\text{rank}(Ω)-1$. More precisely, if either $Ω$ and $Ω'$ have the same type or $Ω$ is of type~III and $Ω'$ is of type~I, then up to automorphisms, $f$ is of the form $f=\imath\circ F$, where $F = F_1\times F_2\colon Ω\to Ω_1'\times Ω_2'$. Here $Ω_1'$, $Ω_2'$ are bounded symmetric domains, the map $F_1\colon Ω\to Ω_1'$ is a standard embedding, $F_2: Ω\to Ω_2'$, and $\imath\colon Ω'_1\times Ω'_2 \to Ω'$ is a totally geodesic holomorphic isometric embedding. Moreover we show that, under the rank condition above, there exists no proper holomorphic map $f: Ω\to Ω'$ if $Ω$ is of type~I and $Ω'$ is of type~III, or $Ω$ is of type~II and $Ω'$ is either of type~I or III. By considering boundary values of proper holomorphic maps on maximal boundary components of $Ω$, we construct rational maps between moduli spaces of subgrassmannians of compact duals of $Ω$ and $Ω'$, and induced CR-maps between CR-hypersurfaces of mixed signature, thereby forcing the moduli map to satisfy strong local differential-geometric constraints (or that such moduli maps do not exist), and complete the proofs from rigidity results on geometric substructures modeled on certain admissible pairs of rational homogeneous spaces of Picard number 1.

Proper holomorphic maps between bounded symmetric domains with small rank differences

TL;DR

This work proves rigidity for proper holomorphic maps between irreducible bounded symmetric domains with small rank differences. By combining boundary analysis, moduli spaces of subgrassmannians, VMRT/CR geometry, and rigidity results for admissible pairs of subvarieties, it derives a canonical factorization: with where is a standard embedding and a totally geodesic isometric embedding, and further identifies cases with nonexistence. The approach hinges on induced moduli maps, their rational extensions, and a sequence of rigidity steps (subgrassmannian respect, induced moduli map rigidity) that culminate in a global diagonalization of . These results connect geometric structure theory with boundary behavior of holomorphic maps, offering a robust framework for rigidity in the setting of Hermitian symmetric domains and their compact duals.

Abstract

In this paper we study the rigidity of proper holomorphic maps between irreducible bounded symmetric domains and with small rank differences: . More precisely, if either and have the same type or is of type~III and is of type~I, then up to automorphisms, is of the form , where . Here , are bounded symmetric domains, the map is a standard embedding, , and is a totally geodesic holomorphic isometric embedding. Moreover we show that, under the rank condition above, there exists no proper holomorphic map if is of type~I and is of type~III, or is of type~II and is either of type~I or III. By considering boundary values of proper holomorphic maps on maximal boundary components of , we construct rational maps between moduli spaces of subgrassmannians of compact duals of and , and induced CR-maps between CR-hypersurfaces of mixed signature, thereby forcing the moduli map to satisfy strong local differential-geometric constraints (or that such moduli maps do not exist), and complete the proofs from rigidity results on geometric substructures modeled on certain admissible pairs of rational homogeneous spaces of Picard number 1.
Paper Structure (14 sections, 38 theorems, 345 equations, 3 tables)

This paper contains 14 sections, 38 theorems, 345 equations, 3 tables.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Hermitian symmetric spaces
  4. Moduli spaces of Hermitian symmetric subspaces and characteristic subspaces
  5. Associated characteristic bundles
  6. Subgrassmannians in the moduli spaces
  7. Rigidity of the pair $(SGr(q,\mathbb C^{2n}), Gr(q, \mathbb C^{2n})$)
  8. Rigidity of subgrassmannian respecting holomorphic maps
  9. Induced moduli map
  10. Rigidity of the induced moduli map
  11. Proof of Theorems
  12. Proof of Theorem \ref{['main']}
  13. Proof of Theorem \ref{['nonexistence']}
  14. Appendix

Key Result

Theorem 1.2

Let $\Omega$ and $\Omega'$ be irreducible bounded symmetric domains of rank $q$ and $q'$, respectively. Suppose Suppose further that either $(1)$$\Omega$ and $\Omega'$ are of the same type or $(2)$$\Omega$ is of type III and $\Omega'$ is of type I. Then, up to automorphisms of $\Omega$ and $\Omega'$, every proper holomorphic map $f\colon \Omega\to \Omega'$ is of the form $f = \imath\circ F$, wher

Theorems & Definitions (75)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 65 more