Abstract

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.

1 Introduction

The object of this paper are the asymptotics of Szegö and Toeplitz operators in a non-standard version of the Berezin–Toeplitz quantization of a complex projective Kähler manifold (M, J, ω).

In Berezin–Toeplitz quantization, one typically adopts as “quantum spaces” the Hermitian spaces H0 (M, A⊗k of global holomorphic sections of higher powers of the polarizing line bundle (A, h); here (A, h) is a positive, hence ample, Hermitian holomorphic line bundle on M. Quantum observables, on the contrary, correspond to Toeplitz operators associated with real C∞ functions on M.

Here we assume given a Hamiltonian action μM of the circle group U(1) = T1 on M, with positive moment map Φ, and admitting a metric-preserving linearization to A. It is then natural to replace the spaces H0 (M, A⊗k with certain new “quantum spaces” which arise by decomposing the Hardy space associated with A into isotypes for the action; these are generally not spaces of sections of powers of A. One is thus led to consider analogues of the usual constructs of Berezin–Toeplitz quantization. In particular, it is interesting to investigate how the symplectic geometry of the underlying action, encapsulated in Φ, influences the semiclassical asymptotics in this quantizationscheme.