Semi-classical asymptotics of partial Bergman kernels on $\mathbb{R}$-symmetric complex manifolds with boundary

Let $M$ be a relatively compact connected open subset with smooth connected boundary of a complex manifold $M^{\mathrm{\prime }}$. Let $(L,h^L)\to M^{\mathrm{\prime }}$ be a positive line bundle over $M^{\mathrm{\prime }}$. Suppose that $M^{\mathrm{\prime }}$ admits a holomorphic $\mathbb{R}$-action which preserves the boundary of $M$ and lifts to $L$. We establish the asymptotic expansion of a partial Bergman kernel associated to a package of Fourier modes of high frequency with respect to the $\mathbb{R}$-action in the high powers of $L$. As an application, we establish an $\mathbb{R}$-equivariant analogue of Fefferman’s and Bell-Ligocka’s result about smooth extension up to the boundary of biholomorphic maps between weakly pseudoconvex domains in ${\mathbb{C}}^n$. Another application concerns the embedding of pseudoconcave manifolds.