Interface asymptotics of partial Bergman kernels on $S^1$-symmetric Kähler manifolds

This article is concerned with asymptotics of equivariant Bergman kernels and partial Bergman kernels for polarized projective Kähler manifolds invariant under a Hamiltonian holomorphic $S^1$ action. Asymptotics of partial Bergman kernel are obtained in the allowed region $\mathcal{A}$ resp. forbidden region $\mathcal{F}$, generalizing results of Shiffman–Zelditch, Shiffman–Tate–Zelditch and Pokorny–Singer for toric Kähler manifolds. The main result gives scaling asymptotics of equivariant Bergman kernels and partial Bergman kernels in the transition region around the interface $\partial \mathcal{A}$, generalizing recent work of Ross–Singer on partial Bergman kernels, and refining the Ross–Singer transition asymptotics to apply to equivariant Bergman kernels.

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Research partially supported by NSF grant DMS-1541126 and by the Stefan Bergman trust.

Received 3 May 2016

Accepted 25 July 2018

Published 9 September 2019