Complex Structures on Product Manifolds

TL;DR

This paper investigates conditions under which the product of two K"ahler manifolds admits an integrable almost complex structure, providing explicit charts in the integrable case and exploring applications to specific complex manifolds.

Contribution

It introduces a new almost complex structure on product manifolds derived from K"ahler quotients and characterizes its integrability, with explicit constructions and applications.

Findings

Derived conditions for integrability of the product almost complex structure.

Constructed explicit holomorphic charts in the integrable case.

Applied results to complex Stiefel manifolds and infinite Calabi-Eckmann manifolds.

Abstract

Let $M_i$, for $i=1,2$, be a K\"ahler manifold, and let $G$ be a Lie group acting on $M_i$ by K\"ahler isometries. Suppose that the action admits a momentum map $\mu_i$ and let $N_i:=\mu_i^{-1}(0)$ be a regular level set. When the action of $G$ on $N_i$ is proper and free, the Meyer--Marsden--Weinstein quotient $P_i:=N_i/G$ is a K\"ahler manifold and $\pi_i:N_i\to P_i$ is a principal fiber bundle with base $P_i$ and characteristic fiber $G$. In this paper, we define an almost complex structure for the manifold $N_1\times N_2$ and give necessary and sufficient conditions for its integrability. In the integrable case, we find explicit holomorphic charts for $N_1\times N_2$. As applications, we consider a non integrable almost-complex structure on the product of two complex Stiefel manifolds and the infinite Calabi-Eckmann manifolds $\mathbb S^{2n+1}\times S(\mathcal{H})$, for $n\geq 1$, where $S(\mathcal{H})$ denotes the unit sphere of an infinite dimensional Hilbert space $\mathcal{H}$