Manifolds with nonnegative isotropic curvature

Abstract. We prove that if (Mn, 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 Kähler and biholomorphic to CP n2. (iv) (M, g) is quaternionic-Kähler. This is implied by the following result: Let (M2n, g) be a compact, locally irreducible Kähler manifold with nonneg-ative isotropic curvature. Then eitherM 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 S. Brendle and R. Schoen on the Ricci flow. 1