Lower-order asymptotics for szegö and toeplitz kernels under hamiltonian circle actions

We consider a natural variant of Berezin-Toeplitz quantization of compact Kähler manifolds, in the presence of a Hamiltonian circle action lifting to the quantizing line bundle. Assuming that the moment map is positive, we study the diagonal asymptotics of the associated Szegö and Toeplitz operators, and specifically their relation to the moment map and to the geometry of a certain symplectic quotient.When the underlying action is trivial and the moment map is taken to be identically equal to one, this scheme coincides with the usual Berezin-Toeplitz quantization. This continues previous work on neardiagonal scaling asymptotics of equivariant Szegö kernels in the presence of Hamiltonian torus actions. Dedicated to Rob Lazarsfeld on the occasion of his 60th birthday