AI 中文总结
该论文证明了Datar-Mete-Song极小斜率猜想，刻画J斜率半稳定性，引入J零轨迹并证明其解析性，还得到环面情形下J方程弱解的光滑性与凯勒性质。
AI 中文摘要
我们证明了Datar-Mete-Song[DMS]提出的通过极小J斜率刻画J斜率半稳定性的猜想。更准确地说，对于紧凯勒流形X上的半稳定凯勒类对(α,β)，每个大且nef的双有理测试类的斜率至少为拓扑J斜率；而不稳定对则存在斜率严格更小的测试类。我们还引入了半稳定对的J零轨迹，并证明当X是紧凯勒曲面或紧环面凯勒流形时，该轨迹是X的解析子集。在环面不变情形下，我们证明了Murakami[Murakami]给出的J方程弱解在X的稠密大环面(ℂ*)ⁿ上是光滑且凯勒的。
英文摘要
We prove a conjecture of Datar-Mete-Song \cite{DMS} characterizing $J$-slope semi-stability by the minimal $J$-slope. More precisely, for a semi-stable pair of Kähler classes $(α,β)$ on a compact Kähler manifold $X$, every big and nef birational test class has slope at least the topological $J$-slope, whereas an unstable pair admits a test class with strictly smaller slope. We also introduce the $J$-null locus of a semi-stable pair and prove that it is an analytic subset of $X$ if $X$ is a compact Kähler surface or a compact toric Kähler manifold. In the toric invariant case, we show that Murakami's \cite{Murakami} weak solution to the $J$-equation is smooth and Kähler on the dense big torus $(\mathbb{C}^*)^n$ of $X$.
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