论文标题
测试配置和代数奇异性类型的关闭
The closures of test configurations and algebraic singularity types
论文作者
论文摘要
给定带有丰富的线条捆绑$ l $的Kähler歧管$ x $，我们认为$ l^1 $ Geodesic射线的度量空间与第一级Chern类$ C_1（L）$相关。我们表征可以通过充足的测试配置近似的射线。同时，我们还表征了所有奇异性类型的Quasi-Plurisubharmonic函数中代数奇异性类型的关闭，并指出了这两个看似无关的问题之间非常紧密的关系。通过Bonavero的全体形态摩尔斯摩尔斯摩尔斯摩尔斯羊皮不平等，代数奇异性类型的算术和非荧光体积重合。我们表明，通常，算术体积占主导地位，而相等性完全符合代数奇异性类型的关闭。类似地，我们对蒙奇（Monge）估计了一般$ l^1 $射线的蒙奇能量。平等完全适用于通过测试配置近似的射线。在两种环境中都给出了其他各种的共同体学和潜在理论特征。作为申请，我们在渐近扩张方面为非架构的蒙加蒙格提供了混凝土公式 - ampère的能量，并显示了投影图的连续性，从$ l^1 $射线到非Archimedean射线。我们还表明，充足的测试配置和过滤的闭合提供了相同的射线。
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.
