Submanifolds of Hermitian Symmetric Spaces

Abstract

We study the problem of non-relativity for a complex Euclidean space and a bounded symmetric domain equipped with their canonical metrics. In particular, we answer a question raised by Di Scala.This paper is dedicated to the memory of Salah Baouendi, a great teacher and a close friend to many of us.

Kähler submanifolds of the symmetrized polydisc

“…P (z, X) is said to be the annihilating function of F (z). Huang and Yuan [14] obtained the following result.…”

“…Therefore, associating Umehara's main result in [21] with the Nakagawa-Takagi embedding, it is easy to get that complex Euclidean spaces and Hermitian symmetric spaces of compact types are not relatives. What is more, Huang and Yuan [14] solved the problem of the non-relativity between complex Euclidean spaces and the bounded symmetric domains by using the properties of Nash algebraic function.…”

“…Our proof uses the idea developed in the work of Huang and Yuan [14], but due to the fact that the nonhomogeneous of G n (n ≥ 2), we cannot assume that G(0) = 0 just as their proof without loss of generality. On the other hand, although that some Cartan-Hartogs domains, researched by Cheng and Niu [7] with the same problem, are also inhomogeneous, they [7] stressed the extra condition G(0) = 0 there.…”

Holomorphic maps from the complex unit ball to Type IV classical domains

“…on U instead of (3) for positive constants λ l . The difference in the proof is that the algebraicity in this case follows from the argument in [HY1,HY2].…”

Kähler submanifolds and the Umehara algebra

“…In addition, the fact that Euclidean spaces and Hermitian symmetric spaces of compact types are not relatives follows from Umehara's result [17] and the classical Nakagawa-Takagi embedding of Hermitian symmetric spaces of compact type into complex projective spaces. Finally, it was shown in [11] that Euclidean spaces and Hermitian symmetric spaces of non-compact types are not relatives.…”