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Semi-classical asymptotics of partial Bergman kernels on $\mathbb{R}$-symmetric complex manifolds with boundary
- Publication Year :
- 2022
-
Abstract
- Let $M$ be a relatively compact connected open subset with smooth connected boundary of a complex manifold $M'$. Let $(L,h^L)\rightarrow M'$ be a positive line bundle over $M'$. Suppose that $M'$ 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.<br />Comment: 55 pages; revised version, new results about smooth extension up to the boundary of biholomorphic maps between weakly pseudoconvex domains with $\mathbb{R}$-action in $\mathbb{C}^n$ added
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2208.12412
- Document Type :
- Working Paper