Optimal pinching for the holomorphic sectional curvature of Hitchin's metrics on Hirzebruch surfaces

The main result of this note is that, for each \(n\in \{1,2,3,\ldots\}\), there exists a Hodge metric on the \(n\)-th Hirzebruch surface whose positive holomorphic sectional curvature is \(\frac{1}{(1+2n)^2}\)-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.