A hyperplane restriction theorem for holomorphic mappings and its application for the gap conjecture

TL;DR

This paper proves a hyperplane restriction theorem for local holomorphic maps between projective spaces, leading to new results on the existence of gaps in rational proper maps between complex balls, applicable to all generalized balls.

Contribution

It introduces a novel hyperplane restriction theorem inspired by Green's theorem, providing the first proof of gaps in rational proper maps for all generalized balls.

Findings

Established a hyperplane restriction theorem for holomorphic mappings.

Proved the existence of gaps in rational proper maps between complex balls.

Demonstrated the phenomenon for all generalized balls.

Abstract

We established a hyperplane restriction theorem for the local holomorphic mappings between projective spaces, which is inspired by the corresponding theorem of Green for homogeneous ideals in polynomial rings. Our theorem allows us to give the first proof for the existence of gaps (albeit smaller) at all level for the rational proper maps between complex unit balls, conjectured by Huang-Ji-Yin. In addition, our proof does not distinguish the unit balls from other generalized balls and thus it simultaneously demonstrates the same phenomenon for all generalized balls.