Toric partial density functions and stability of toric varieties

TL;DR

This paper develops a distributional expansion for partial density functions on toric Kähler manifolds and uses it to connect constant scalar curvature metrics with slope semi-stability, impacting the study of toric K-stability.

Contribution

It introduces a distributional expansion for partial density functions and applies it to establish a link between scalar curvature and slope semi-stability in toric varieties.

Findings

Distributional expansion of partial density functions as k→∞

Constant scalar curvature implies slope semi-stability

Applications to toric K-stability

Abstract

Let $(L, h)\to (X, \omega)$ denote a polarized toric K\"ahler 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.