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We established a hyperplane restriction theorem for the local holomorphicmappings between projective spaces, which is inspired by the correspondingtheorem of Green for homogeneous ideals in polynomial rings. Our theorem allowsus to give the first proof for the existence of gaps (albeit smaller) at alllevel for the rational proper maps between complex unit balls, conjectured byHuang-Ji-Yin. In addition, our proof does not distinguish the unit balls fromother generalized balls and thus it simultaneously demonstrates the samephenomenon for all generalized balls.

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