Complex structures on product manifolds

Abstract

Let Mi{M}_{i}, for i=1i=1, 2, be a Kähler manifold, and let GG be a compact Lie group acting on Mi{M}_{i} by Kähler isometries. Suppose that the action admits a momentum map μi{\mu }_{i}, and let Ni≔μi−1(0){N}_{i}:= {\mu }_{i}^{-1}\left(0) be a regular-level set. When the action of GG on Ni{N}_{i} is proper and free, the Meyer-Marsden-Weinstein quotient Pi≔Ni∕G{P}_{i}:= {N}_{i}/G is a Kähler manifold and πi:Ni→Pi{\pi }_{i}:{N}_{i}\to {P}_{i} is a principal fiber bundle with base Pi{P}_{i} and characteristic fiber GG. In this article, we define an almost-complex structure on the manifold N1×N2{N}_{1}\times {N}_{2} and give necessary and sufficient conditions for its integrability. In the integrable case, we find explicit holomorphic charts for N1×N2{N}_{1}\times {N}_{2}. As applications, we consider a nonintegrable almost-complex structure on the product of two complex Stiefel manifolds and the infinite Calabi-Eckmann manifolds S2n+1×S(ℋ){{\mathbb{S}}}^{2n+1}\times S\left({\mathcal{ {\mathcal H} }}), for n≥1n\ge 1, where S(ℋ)S\left({\mathcal{ {\mathcal H} }}) denotes the unit sphere of an infinite-dimensional complex Hilbert space ℋ{\mathcal{ {\mathcal H} }}.

Keywords

• group action
• momentum map
• reduction
• complex structures
• primary 53c55
• 57s20