Under the conditions K_M=O(-2D), D a smooth Calabi-Yau, and O_{3D}(D) trivial, the compact Kähler manifold M is biholomorphic to P^1 times D.
The splitting theorem for manifolds of non-negative Ricci curvature
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A Holomorphic Splitting Theorem
Under the conditions K_M=O(-2D), D a smooth Calabi-Yau, and O_{3D}(D) trivial, the compact Kähler manifold M is biholomorphic to P^1 times D.