Abstract:

Let M be a complex manifold with smooth boundary X, which admits a compact connected Lie group G acting holomorphically and preserving X. We establish a full asymptotic expansion for the G-invariant Bergman kernel under certain assumptions. As an application, we get G-invariant version of Fefferman's result about regularity of biholomorphic maps on strongly pseudoconvex domains of $C^n$. Moreover, we show that the Guillemin–Sternberg map on a complex manifold with boundary is Fredholm by developing reduction to boundary technique, which establishes “quantization commutes with reduction” in this case, as an analogue of its CR version. This is a joint work with Chin-Yu Hsiao, Xiaoshan Li and Guokuan Shao. (Mathematische Annalen, https://doi.org/10.1007/s00208-024-02865-1)