复曲面部分密度函数与复曲面簇的稳定性
Toric partial density functions and stability of toric varieties
摘要：
令$(L, h)\to (X, ω)$表示一个极化复曲面Kähler流形。固定一个环面子流形$Y$，用$\hatρ_{tk}:X\to \mathbb{R}$表示偏密度函数，该偏密度函数对应于将$L^k$的光滑截面投影到$L^k$的全纯截面上的偏Bergman核，该全纯截面至少沿$Y$为$tk$，对于固定的$t>0$，使得$tk\in \mathbb{N}$。我们证明了$\hatρ_{tk}$作为$k\to \infty$的分布展开的存在性，包括作为$X$上的分布的$k^{n-1}$的系数的识别。这个展开式被用来给出一个直接的证明：如果$ω$有常数量曲率，那么$(X, L)$一定是关于$Y$的斜率半稳定的。对于更一般的部分密度函数也得到了类似的结果。这些结果对于研究复曲面簇的复曲面K-稳定性具有类似的应用。
英文摘要：
Let $(L, h)\to (X, ω)$ denote a polarized toric Kähler manifold. Fix a toric submanifold $Y$ and denote by $\hatρ_{tk}:X\to \mathbb{R}$ 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 $\hatρ_{tk}$ 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 $ω$ 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.
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