Toric partial density functions and stability of toric varieties

Abstract

Let $(L,h)\to (X,\omega )$ denote a polarized toric Kähler manifold. Fix a toric submanifold $Y$ and denote by ${\hat{\rho }}_{tk}:X\to \mathbb{R}$ the partial density function corresponding to the partial Bergman kernel projecting smooth sections of $L^k$ onto holomorphic sections of $L^k$ that vanish to order at least $tk$ along $Y$, for fixed $t>0$ such that $tk\in \mathbb{N}$. We prove the existence of a distributional expansion of ${\hat{\rho }}_{tk}$ as $k\to \infty$, including the identification of the coefficient of $k^{n-1}$ as a distribution on $X$. This expansion is used to give a direct proof that if $\omega$ 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.

Accepted by Mathematische Annalen on 13 September 2013

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• K-stability and Kähler-Einstein metrics
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• Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities
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