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Let $M$ be a relatively compact connected open subset with smooth connectedboundary of a complex manifold $M'$. Let $(L,h^L)\rightarrow M'$ be a positiveline bundle over $M'$. Suppose that $M'$ admits a holomorphic$\mathbb{R}$-action which preserves the boundary of $M$ and lifts to $L$. Weestablish the asymptotic expansion of a partial Bergman kernel associated to apackage of Fourier modes of high frequency with respect to the$\mathbb{R}$-action in the high powers of $L$. As an application, we establishan $\mathbb{R}$-equivariant analogue of Fefferman's and Bell-Ligocka's resultabout smooth extension up to the boundary of biholomorphic maps between weaklypseudoconvex domains in $\mathbb{C}^n$. Another application concerns theembedding of pseudoconcave manifolds.

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