[论文解读] The Kähler submanifolds between the ball bundles and the complex Euclidean space

Mingming Chen, Yihong Hao+1 more|arXiv (Cornell University)|Jan 1, 2026

Geometry and complex manifolds被引用 0

一句话总结

本论文给出一个充分条件（通过指数Nash-代数Kähler流形与有理指数势函数），确保复杂欧几里得空间与某些 Hermitian 向量丛的球面束之间不存在共有的Kähler子流形；并给出 Bergman 与 Kähler–Einstein 球面束在特定基底空间上的非存在性（以及某些存在性）结果。

ABSTRACT

In this paper, we provide a sufficient condition on the non-existence of the common Kähler submanifolds between the complex Euclidean space and the ball bundles of some Hermitian vector bundles over Kähler manifolds. Then we get the non-existence theorems on several classes of ball bundles whose base spaces are Hermitian symmetric spaces or the complete Kähler-Einstein manifolds.

研究动机与目标

• Motivate and study the relativity problem (common Kähler submanifolds) between complex Euclidean space and ball bundles over Kähler manifolds.
• Introduce exponent Nash-algebraic (and exponent Nash-algebraic) Kähler manifolds and a framework to test non-existence of common submanifolds.
• Prove a general non-existence theorem (Theorem 1) for holomorphic isometric embeddings under the Nash-algebraic framework.
• Apply the main theorem to derive non-existence results for several classes of ball bundles with Bergman metrics and a KE metric.
• Discuss how the conditions A (Nash-algebraic base) and B (rational exp(u)) prevent relative submanifolds.

提出的方法

• Define a Kähler ansatz for the ball bundle metric ω_{B(E_k)} involving Ricci and base Kähler forms plus a potential term u.
• Introduce exponent Nash-algebraic (Definition 1) and use polarization, Nash algebraic properties, and composition rules (Lemmas 1–4) to control functional relations.
• Derive a fundamental equation F^*ω_{C^n} = μ G^*ω_{B(E_k)} on a domain V and show that under the hypotheses F must be constant (Theorem 1).
• Utilize polarization of Kähler potentials and Nash-algebraic function theory (Lemmas 2–5) to preclude non-constant holomorphic maps from V to C^n given the structure of ω_{B(E_k)}.
• Apply the theorem to specific ball bundles with Bergman or KE metrics to obtain corollaries asserting non-existence (and some existence) results.

实验结果

研究问题

• RQ1 When can a common Kähler submanifold exist between complex Euclidean space and ball bundles over Hermitian vector bundles?
• RQ2 What conditions on the base Kähler manifold and the potential function u rule out non-constant holomorphic isometries into ball bundles?
• RQ3 How do Nash-algebraic or exponent Nash-algebraic properties of the base metric influence relativity with C^n?
• RQ4 Can the main non-existence theorem be specialized to Bergman ball bundles over symmetric or KE bases to obtain concrete non-relativity results?
• RQ5 What specific corollaries follow for ball bundles over compact Hermitian symmetric spaces, bounded symmetric domains, and complete KE manifolds?

主要发现

• A general non-existence theorem (Theorem 1) showing that non-constant holomorphic maps into Bergman ball bundles are ruled out when the base is exponent Nash-algebraic and exp(u) is rational.
• Under the ansatz for ω_{B(E_k)}, if the base metric satisfies the exponent Nash-algebraic condition and exp(u) is rational, any holomorphic map F into C^n paired with G into the ball bundle must be constant.
• Theorem 1 implies corollaries that yield non-existence results for common Kähler submanifolds between complex Euclidean space and several ball bundles over Hermitian symmetric spaces (via Bergman metrics).
• Corollaries 1–4 provide explicit non-existence statements for ball bundles in top exterior power bundles, Hartogs domains over bounded symmetric domains, and Hartogs domains over complete flat spaces, as well as negative KE base manifolds.]