A compact Kähler manifold with positive Ricci curvature has volume at most 2 n^n/(n+1)^n times the projective space volume, unless it is the quadric or P^1 times P^{n-1}.
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The sharp volume gap for K\"ahler manifolds with positive Ricci curvature
A compact Kähler manifold with positive Ricci curvature has volume at most 2 n^n/(n+1)^n times the projective space volume, unless it is the quadric or P^1 times P^{n-1}.