Mathematics – Spectral Theory
Scientific paper
2007-06-20
Mathematics
Spectral Theory
16 pages
Scientific paper
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.
Burns David
Guillemin Victor
Uribe Alejandro
No associations
LandOfFree
The spectral density function of a toric variety does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The spectral density function of a toric variety, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The spectral density function of a toric variety will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-34759