We know that the Grassmannian manifold $G(k,\mathbb{C}^n)$ can be embedded in the projective space $\mathbb{C}P^N$ for $N= {n\choose k}-1$ by the Plucker embedding $P$.
On $\mathbb{C}P^N$ we have the hermitian metric $h_a=\frac{|z_a|^2}{|z_0|^2+\cdots+|z_N|^2}$ over $U_a=\{z_a\neq0\}$ defined on the line bundle $\mathcal{O}(1).$ See related question here.
Suppose we consider $P^* (\mathcal{O}(1))$ over the Grassmannian. How does the Hermitian metric look like here?
Any help/hints/ideas will be appreciated!
