On geometric properties of holomorphic isometries between bounded symmetric domains

TL;DR

This paper investigates the geometric properties of holomorphic isometries between bounded symmetric domains, revealing how their images relate to affine-linear structures and constructing specific submersions to understand their global behavior.

Contribution

It introduces new geometric insights into holomorphic isometries, including affine-linear intersection properties and a construction of a holomorphic submersion linking the domains.

Findings

Images of affine-linear sections are intersections with affine-linear subspaces.

Constructed a surjective holomorphic submersion related to the isometries.

Derived geometric properties using classical complex-analytic results.

Abstract

We study holomorphic isometries between bounded symmetric domains with respect to the Bergman metrics up to a normalizing constant. In particular, we first consider a holomorphic isometry from the complex unit ball into an irreducible bounded symmetric domain with respect to the Bergman metrics. In this direction, we show that images of (nonempty) affine-linear sections of the complex unit ball must be the intersections of the image of the holomorphic isometry with certain affine-linear subspaces. We also construct a surjective holomorphic submersion from a certain subdomain of the target bounded symmetric domain onto the complex unit ball such that the image of the holomorphic isometry lies inside the subdomain and the holomorphic isometry is a global holomorphic section of the holomorphic submersion. This construction could be generalized to any holomorphic isometry between bounded symmetric domains with respect to the \emph{canonical K\"ahler metrics}. Using some classical results for complex-analytic subvarieties of Stein manifolds, we have obtained further geometric results for images of such holomorphic isometries.