极化Kähler流形上的判别式和高阶K-能量
Discriminants and Higher K-energies on Polarized Kähler Manifolds
摘要：
给定一个紧致极化Kähler流形$X\hookrightarrow\mathbb{CP}^N$，由$\mathrm{SL}(N+1,\mathbb{C})$参数化的$X\hookrightarrow\mathbb{CP}^N$上的Bergman度量空间对应于Kähler类中Kähler势空间中的一个稠密集$N\to\infty$。定义在Kähler类上的$k$ th K-能量泛函的临界点对应于具有调和$k$ th Chern形式的度量.在本文中，它表明，较高的K-能量泛函，当限制到伯格曼度量，是表示为能量的某些对向量（张量产品的判别式）。相应地，我们得到了这些泛函沿单参数子群的渐近行为及其有界性的结果。
英文摘要：
Given a compact polarized Kähler manifold $X\hookrightarrow\mathbb{CP}^N$, the space of Bergman metrics on $X$, parameterized by $\mathrm{SL}(N+1,\mathbb{C})$, corresponds to a dense set in the space of Kähler potentials in the Kähler class as $N\to\infty$. Critical points of the $k$th K-energy functional, which is defined on the Kähler class, correspond to metrics with harmonic $k$th Chern form. In this paper it is shown that the higher K-energy functionals, when restricted to the Bergman metrics, are expressible as the energies of certain pairs of vectors (tensors products of discriminants). Consequentially, we obtain results on the asymptotic behavior of these functionals along 1-parameter subgroups and their boundedness properties.
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