Orthogonal pair and a rigidity problem for Segre maps between hyperquadrics

TL;DR

This paper investigates orthogonal pairs between projective spaces with Hermitian forms, revealing rigidity properties of holomorphic orthogonal pairs and deriving a rigidity theorem for Segre maps of hyperquadrics.

Contribution

It introduces orthogonal pairs as generalizations of Segre maps and establishes their rigidity under certain conditions, extending previous work on holomorphic maps between hyperquadrics.

Findings

Non-degenerate holomorphic orthogonal pairs exhibit rigidity.

A new rigidity theorem for Segre maps related to Heisenberg hypersurfaces.

Extension of orthogonal map theory to degenerate Hermitian forms.

Abstract

Being motivated by the orthogonal maps studied in \cite{GN1}, orthogonal pairs between the projective spaces equipped with possibly degenerate Hermitian forms were introduced. In addition, orthogonal pairs are generalizations of holomorphic Segre maps between Segre families of real hyperquadrics. We showed that non-degenerate holomorphic orthogonal pairs also have certain rigidity properties under a necessary codimension restriction. As an application, we got a rigidity theorem for Segre maps related to Heisenberg hypersurfaces obtained by Zhang in \cite{Zh}.