Off-diagonal estimates of partial Bergman kernels on $S^1$-symmetric Kähler manifolds

We establish local asymptotic estimates of partial Bergman kernels on closed, $S^1$-symmetric Kähler manifolds. The main result concerns the scaling asymptotics of partial Bergman kernels at generic off-diagonal points in which they are not negligible. The case of the two-dimensional sphere is discussed in detail.