Complex structures on product manifolds

Let M_i , for i=1 , 2, be a Kahler manifold, and let G G be a compact Lie group acting on M-i by Kahler isometries. Suppose that the action admits a momentum map mu(i) , and let N-i := mu(-1)(i) (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 Kahler manifold and pi(i) : N-i -> P-i is a principal fiber bundle with base P-i and characteristic fiber G . In this article, we define an almost-complex structure on the manifold N-1 x N-2 and give necessary and sufficient conditions for its integrability. In the integrable case, we find explicit holomorphic charts for N-1 x 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 x S (H), for n > 1 , where S (H) denotes the unit sphere of an infinite-dimensional complex Hilbert space H.