Kähler-Einstein度规与体积极小化
Kähler-Einstein metrics and volume minimization
摘要：
证明了：若$\mathbb{Q}$ -Fano簇$V$特殊退化为Kähler-Einstein $\mathbb{Q}$ -Fano簇$V$，则对于任意具有$r\in \mathbb{Q}_{>0}$的充足Cartier因子$H=-r^{-1} K_V$，在以仿射锥$\mathcal{C}:=C(V,H)$顶点为中心的所有实赋值中，在典范赋值${\rm ord}_V$处，归一化体积$\widehat{\rm vol}(v)=A_{\mathcal{C}}^n(v)\cdot {\rm vol}(v)$全局最小.这也被推广到对数和轨道设置。因此，我们完成了[arXiv：1511.08164]中关于任何光滑Fano流形的K-半稳定性的等价刻画的猜想的确认。我们还证明了与光滑的Sasaki-Einstein度量的Reeb向量场相关的估值在相应的Kähler锥上最小化了$\widehat{\rm vol}$。这些结果加强了Martelli-Sparks-Yau的最小化结果[Martelli et al 08]。
英文摘要：
We prove that if a $\mathbb{Q}$-Fano variety $V$ specially degenerates to a Kähler-Einstein $\mathbb{Q}$-Fano variety $V$, then for any ample Cartier divisor $H=-r^{-1} K_V$ with $r\in \mathbb{Q}_{>0}$, the normalized volume $\widehat{\rm vol}(v)=A_{\mathcal{C}}^n(v)\cdot {\rm vol}(v)$ is globally minimized at the canonical valuation ${\rm ord}_V$ among all real valuations which are centered at the vertex of the affine cone $\mathcal{C}:=C(V,H)$. This is also generalized to the logarithmic and the orbifold setting. As a consequence, we complete the confirmation of a conjecture in [arXiv:1511.08164] on an equivalent characterization of K-semistability for any smooth Fano manifold. We also prove that the valuation associated to the Reeb vector field of a smooth Sasaki-Einstein metric minimizes $\widehat{\rm vol}$ over the corresponding Kähler cone. These results strengthen the minimization result of Martelli-Sparks-Yau [Martelli et al 08].
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