Toric partial density functions and stability of toric varieties

الأصل · EN

Let (L, h)→ (X, ω) denote a polarized toric Kähler manifold. Fix a toric submanifold Y and denote by ρtk:X→ R the partial density function corresponding to the partial Bergman kernel projecting smooth sections of Lᵏ onto holomorphic sections of Lᵏ that vanish to order at least tk along Y, for fixed t>0 such that tk∈ N. We prove the existence of a distributional expansion of ρtk as k→ ∞, including the identification of the coefficient of kⁿ⁻¹ as a distribution on X. This expansion is used to give a direct proof that if ω has constant scalar curvature, then (X, L) must be slope semi-stable with respect to Y. 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.