Can I read Rationality of Proper Holomorphic Maps Between Bounded Symmetric Domains of the First Kind on EtoBox?

Rationality of Proper Holomorphic Maps Between Bounded Symmetric Domains of the First Kind by Kim, Sung-Yeon is a scholarly article available to read on EtoBox.

What is Rationality of Proper Holomorphic Maps Between Bounded Symmetric Domains of the First Kind about?

Let $D_{p,q}$ and $D_{p',q'}$ be irreducible bounded symmetric domains of the first kind with rank $q$ and $q'$, respectively and let $f:D_{p,q}\to D_{p',q'}$ be a proper holomorphic map that extends $C^2$ up to the boundary. In this paper we show that if $q, q'\geq 2$ and $f$ maps Shilov boundary of $D_{p,q}$ to Shilov boundary of $D_{p',q'}$, then $f$ is of the form $f = \imath\circ F$, where $$ F=F_1\times F_2\colon D_{p,q}\to \Omega_1'\times \Omega_2', $$ $\Omega_1'$ and $\Omega_2'$ are bounded symmetric domains, $F_1 \colon D_{p,q}\to \Omega_1'$ is a proper rational map, $F_2:D_{p,q}\to \Omega_2'$ is not proper and $\imath: \Omega_1' \times \Omega'_2 \hookrightarrow D_{p',q'}$ is a holomorphic totally geodesic isometric embedding of a reducible bounded symmetric domain $\Omega_1' \times \Omega_2'$ into $D_{p',q'}$ with respect to canonical K\"ahler-Einstein metrics. Moreover, if $p>q$, then $f$ is a rational map. As an application we show that a proper holomorphic map $f:D_{p,q}\to D_{p',q'}$ that extends $C^\infty$ up to the boundary is a rational map or a totally geodesic isometric embedding with respect to the Kobayashi metrics, if $3\leq q \leq q'\leq 2q-1.$

Author

Kim, Sung-Yeon

Published

2023

Language

EN