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Toric partial density functions and stability of toric varieties
Let denote a polarized toric K\"ahler manifold. Fix a
toric submanifold and denote by the
partial density function corresponding to the partial Bergman kernel projecting
smooth sections of onto holomorphic sections of that vanish to
order at least along , for fixed such that . We
prove the existence of a distributional expansion of as , including the identification of the coefficient of as a
distribution on . This expansion is used to give a direct proof that if
has constant scalar curvature, then must be slope semi-stable
with respect to . Similar results are also obtained for more general partial
density functions. These results have analogous applications to the study of
toric K-stability of toric varieties.Comment: Accepted by Mathematische Annalen on 13 September 201
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