Compact, connected, complex manifolds that admit a compact transitive group of holomorphic automorphisms
Lei Ni, Nolan Wallach
TL;DR
This work develops a Lie algebraic approach to compact homogeneous complex manifolds with a transitive compact group of biholomorphisms, introducing a canonical abelian subalgebra attached to an integrable complex structure on $\mathfrak{g}/\mathfrak{h}$ and linking it to the Nijenhuis tensor $N$. It shows how integrability and the existence of invariant complex structures can be characterized and constructed via centralizers of abelian subalgebras and parabolic subalgebras $\mathfrak{p}\subset\mathfrak{g}_{\mathbb{C}}$, enabling a classification of $\mathfrak{g}/\mathfrak{h}$-complex structures through $J(\mathfrak{p},J_1)$. The results generalize and streamline classical theorems of Wang and Tits, recovering them without topological hypotheses and revealing that invariant complex structures arise from a parabolic data, yielding a holomorphic fibration $G/H\to G/M$ with base a flag variety and fiber a complex torus. In the irreducible case, the framework identifies Hermitian symmetric pairs of compact type, aligning with Wolf–Ziller classifications and underscoring the geometric structure of compact Hermitian homogeneous manifolds.
Abstract
The purpose of this paper is to develop a Lie algebraic approach to obtain new proofs of important results of H.-C. Wang, Tits and Wolf-Wang-Ziller on compact complex homogeneous manifolds emphasizing only those that admit a transitive compact group of biholomorphic transformations. The method only uses some standard results in Lie theory. The new approach provides a method of associating a canonical abelian Lie algebra with a given integrable complex structure on a compact Lie algebra which extends the earlier work of Samelson and Pittie.