测试配置的闭包与代数奇点类型
The closures of test configurations and algebraic singularity types
AI总结：
本文研究了 Kähler 流形上 $L^1$ 测地射线空间，刻画了可由测试配置逼近的射线与代数奇点类型的闭包，揭示了二者的紧密联系，并给出了体积与能量估计及非阿基米德 Monge--Ampère 能量的应用。
摘要：
给定具有丰富线丛 $L$ 的 Kähler 流形 $X$，我们考虑与第一陈类 $c_1(L)$ 相关的 $L^1$ 测地射线度量空间。我们刻画了可由丰富测试配置逼近的射线。同时，我们也刻画了拟多次下调和函数所有奇点类型中代数奇点类型的闭包，指出了这两个看似无关问题之间极其密切的关系。根据 Bonavero 的全纯 Morse 不等式，代数奇点类型的算术体积与非极化体积相等。我们证明在一般情况下算术体积主导非极化体积，且等式恰好成立于代数奇点类型的闭包上。类似地，我们利用 Legendre 变换给出了任意 $L^1$ 射线 Monge--Ampère 能量的估计式，等式恰好对可由测试配置逼近的射线成立。我们在两种设定下给出了其他各种上同调与位势理论的刻画。作为应用，我们给出了非阿基米德 Monge--Ampère 能量基于渐近展开的具体公式，并证明了从 $L^1$ 射线到非阿基米德射线的投影映射的连续性。我们还证明了丰富测试配置的闭包与滤过给出相同的射线集合。
英文摘要：
Given a Kähler manifold $X$ with an ample line bundle $L$, we consider the metric space of $L^1$ geodesic rays associated to the first Chern class $c_1(L)$. We characterize rays that can be approximated by ample test configurations. At the same time, we also characterize the closure of algebraic singularity types among all singularity types of quasi-plurisubharmonic functions, pointing out the very close relationship between these two seemingly unrelated problems. By Bonavero's holomorphic Morse inequalities, the arithmetic and non-pluripolar volumes of algebraic singularity types coincide. We show that in general the arithmetic volume dominates the non-pluripolar one, and equality holds exactly on the closure of algebraic singularity types. Analogously, we give an estimate for the Monge--Ampère energy of a general $L^1$ ray in terms of the arithmetic volumes along its Legendre transform. Equality holds exactly for rays approximable by test configurations. Various other cohomological and potential theoretic characterizations are given in both settings. As applications, we give a concrete formula for the non-Archimedean Monge--Ampère energy in terms of asymptotic expansion, and show the continuity of the projection map from $L^1$ rays to non-Archimedean rays. We also show that the closure of ample test configurations and filtrations gives the same set of rays.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
