Complex structures on product manifolds

Abstract Let M i M₈, for i = 1 i=1, 2, be a Kähler manifold, and let G G be a compact Lie group acting on M i M₈ by Kähler isometries. Suppose that the action admits a momentum map μ i ₈, and let N i ≔ μ i − 1 (0) N₈: = ₈^-1 (0) be a regular-level set. When the action of G G on N i N₈ is proper and free, the Meyer-Marsden-Weinstein quotient P i ≔ N i ∕ G P₈: = N₈/G is a Kähler manifold and π i: N i → P i ₈: N₈ P₈ is a principal fiber bundle with base P i P₈ and characteristic fiber G G. In this article, we define an almost-complex structure on the manifold N 1 × N 2 N₁ N₂ and give necessary and sufficient conditions for its integrability. In the integrable case, we find explicit holomorphic charts for N 1 × N 2 N₁ N₂. As applications, we consider a nonintegrable almost-complex structure on the product of two complex Stiefel manifolds and the infinite Calabi-Eckmann manifolds S 2 n + 1 × S (ℋ) {S}^2n+1 S ({ H }), for n ≥ 1 n 1, where S (ℋ) S ({ H }) denotes the unit sphere of an infinite-dimensional complex Hilbert space ℋ { H }.