Partial Bergman kernels and determinantal point processes on K\"ahler manifolds

We compute the full off-diagonal asymptotics of the equivariant and partial Bergman kernels associated with a circle action on a prequantized K\"ahler manifold with bounded geometry at infinity, then use these results to compute the asymptotics of the linear statistics of the associated determinantal point process as the number of points grows to infinity, showing that its distribution converges to a centered normal variable with variance given by the sum of an $H^1$-norm squared in the bulk and an $H^{1/2}$-norm squared on the boundary of the associated droplet.