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Harmonic Analysis on Toric Varieties

B. Shiffman·T. Tate·S. Zelditch
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Abstract

Harmonic analysis on a toric Kahler variety M refers to the orthonormal basis of eigenfunctions of the complex torus action on the spaces H^0(M, L^N) of holomorphic sections of powers of a positive line bundle L and the Fourier multipliers that act on them. Using this harmonic analysis, we give an exact formula for the Szego kernel as a Fourier multiplier applied to the pull back of the Szego kernel of projective space under a monomial embedding. The Fourier multiplier involves a partition function of the convex lattice polytope P associated to M. We further prove that this Fourier multiplier is a Toeplitz operator, and as a corollary we obtain an oscillatory integral formula for the characters χ_{NP} of the torus action on H^0(M, L^N).

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B. Shiffman, T. Tate, S. Zelditch. 2003-03-26. Harmonic Analysis on Toric Varieties. https://arxiv.org/abs/math/0303337

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