Let $X$ be a compact Riemann surface of genus at least 2. Let $K$ denote the canonical line bundle, and $E$ be any vector bundle. Let $P^{(m)}$ be the projection map from the space $L^2(X,K^mE)$ of plurisections to the Hardy space $A^2(X,K^mE)$ of holomorphic section. The Bergman kernel $P^{(m)}(z,\bar{w})$ is the smooth integral kernel associated to the projection map $P^{(m)}$.
Since we are on dimension 1, the theorem about asymptotic expansion tells us that there are only two smooth coefficients $b_0^{(m)}$ and $b_1^{(m)}$ such that $P^{(m)}(z,\bar{w})=b_0^{(m)}(z,\bar{w})m+ b_1^{(m)}(z,\bar{w}) + O(1/m)$.
If $(s_i)$ is an orthonormal basis of $A^2(X,K)$, then $b_0^{(m)}(z,\bar{w})=(\sum s_i(z) s_i(\bar{w}))^m$. Along the diagonal, the book by Ma and Marinescu gives the expression for $b_1$. Namely, $b_1=\frac{1}{8\pi}\det (\frac{\dot{R}^K}{2\pi})[r_\omega^X -2\Delta_\omega(\log(\det \dot{R}^K))+4\sqrt{-1}\Omega_\omega(R^E)]$. Does anyone know of the off diagonal expression for $b_1$?
Even better, is there an expression for Bergman kernel itself $P^{(m)}(z,\bar{w})$ itself for this low-dimensional case?