We prove that if (M-n, g), n >= 4, is a compact, orientable, locally irreducible Riemannian manifold with nonnegative isotropic curvature, then one of the following possibilities hold: (i) M admits a metric with positive isotropic curvature. (ii) (M, g) is isometric to a locally symmetric space. (iii) (M, g) is Kahler and biholomorphic to CPn/2. (iv) (M, g) is quaternionic-Kahler. This is implied by the following result: Let (M-2n, g) be a compact, locally irreducible Kahler manifold with nonnegative isotropic curvature. Then either M is biholomorphic to CPn or isometric to a compact Hermitian symmetric space. This answers a question of Micallef and Wang in the affirmative. The proof is based on the recent work of Brendle and Schoen on the Ricci flow.
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