Abstract. The main result of this note is that, for each n ∈ {1, 2, 3, . . .}, there exists a Hodge metric on the n-th Hirzebruch surface whose positive holomorphic sectional curvature is 1 (1+2n) 2 -pinched. The type of metric under consideration was first studied by Hitchin in this context. In order to address the case n = 0, we prove a general result on the pinching of the holomorphic sectional curvature of the product metric on the product of two Hermitian manifolds M and N of positive holomorphic sectional curvature.

Abstract. We generalize a construction of Hitchin to prove that, given any compact Kähler manifold M with positive holomorphic sectional curvature and any holomorphic vector bundle E over M , the projectivized vector bundle P(E) admits a Kähler metric with positive holomorphic sectional curvature.

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