Higher Rank Bergman Kernels on Compact Riemann Surfaces

Let X be a compact Riemann surface equipped with a real-analytic Kähler form $\omega$ and let E be a holomorphic vector bundle over $X$ equipped with a real-analytic Hermitian metric $h$. Suppose that the curvature of $h$ is Griffiths-positive. We prove the existence of a global asymptotic expansion in powers of $k$ of the Bergman kernel associated to $(S^{k}E,S^{k}h)$ and $\omega$.