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

Abstract

This article is concerned with asymptotics of equivariant Bergman kernels and partial Bergman kernels for polarized projective Kahler 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 Kahler 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.