Complex Structures on Product Manifolds
We generalize the construction of a complex structure on a product of two odd-dimensional spheres from a classical paper by Calabi and Eckmann. More precisely, let M_i, for i = 1,2, be a Kähler manifold, and let G be a compact Lie group acting on M_i by Kähler isometries. Suppose that the action admits a momentum map µ_i and assume that its zero-level set N_i is a regular submanifold of M_i. We define an almost complex structure on the manifold N_1 x N_2 and provide necessary and sufficient conditions for its integrability. In the integrable case, we find explicit holomorphic charts for N_1 x N_2. If time permits, we will also present some applications.
This is a joint work with Leonardo Biliotti.
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