The spectral density function of a toric variety

Abstract

For a Kahler manifold (X, \omega) with a holomorphic line bundle L and metric h such that the Chern form of L is \omega, the spectral measures are the measures \mu_N = \sum |s_{N,i}|^2 \nu, where \{s_{N,i}\}_i is an L^2-orthonormal basis for H^0(X, L^{\otimes N}), and \nu is Liouville measure. We study the asymptotics in N of \mu_N for (X, L) a Hamiltonian toric manifold, and give a precise expansion in terms of powers 1/N^j and data on the moment polytope \Delta of the Hamiltonian torus K acting on X. In addition, for an infinitesimal character k of K and the unique unit eigensection s_{Nk} for the character Nk of the torus action on H^0(X, L^N), we give a similar expansion for the measures \mu_{Nk} = |s_{Nk}|^2 \nu. A final remark shows that the eigenbasis \{s_{k}, k \in \Delta \cap \mathbb{Z}^{\dim K} \} is a Bohr-Sommerfeld basis in the sense of Tyurin. Some of the present results are related to work of Shiffman, Tate and Zelditch. The present paper uses no microlocal analysis, but rather an Euler-Maclaurin formula for Delzant polytopes.