算术θ不变量与算术平衡测度
Arithmetic theta invariants and arithmetic equilibrium measures
浏览论文内容
中文总结 AI 辅助
本研究引入算术簇上埃尔米特线丛的两个新算术不变量，导出内蕴算术平衡测度，证明其可检测算术正性、支配算术体积变分，为相关问题提供新的分析框架。
中文摘要 AI 辅助
我们引入了与算术簇上的埃尔米特线丛相关的两个新的算术不变量，它们在算术框架中起到伯格曼畸变函数的作用。作为第一步，我们证明这两个函数是渐近等价的，由此引入了一个内蕴测度$\widehatμ^{\sup}_{\mathrm{eq}}$，并证明它是平衡测度的自然算术类似物。我们证明$\widehatμ^{\sup}_{\mathrm{eq}}$可作为算术正性的检测器，具体而言，我们确定了弱nef埃尔米特线丛对应的该测度，并在环面情形下得到了完全一般的结果。此外，$\widehatμ^{\sup}_{\mathrm{eq}}$支配算术体积函数及其变分性质，更准确地说，我们建立了算术体积及其一阶变分关于$\widehatμ^{\sup}_{\mathrm{eq}}$的弱积分表示，这些公式为Yuan--Zhang的体积导数提供了测度论解释，并揭示了其与复多重位势理论中变分公式的结构相似性。我们的方法与此前基于Harder--Narasimhan滤过或算术Okounkov体的处理方式有根本区别，它将算术体积置于理论核心，表明渐近体积不变量而非高度，为等分布与算术动力学问题提供了自然的分析框架。
英文摘要
We introduce two new arithmetic invariants associated with Hermitian line bundles on arithmetic varieties, which play the role of the Bergman distortion function in the arithmetic setting. As a first step, we show that these two functions are asymptotically equivalent. This leads to the introduction of an intrinsic measure $ \widehatμ^{\sup}_{\mathrm{eq}},$ which we show to be a natural arithmetic analogue of the equilibrium measure. We prove that $\widehatμ^{\sup}_{\mathrm{eq}}$ serves as a detector of arithmetic positivity. In particular, we determine it for weakly nef Hermitian line bundles and, in full generality, in the toric case. Moreover, $\widehatμ^{\sup}_{\mathrm{eq}}$ governs the arithmetic volume function and its variational properties. More precisely, we establish weak integral representations of the arithmetic volume and of its first variation in terms of $\widehatμ^{\sup}_{\mathrm{eq}}$. These formulas provide a measure-theoretic interpretation of the volume derivative of Yuan--Zhang and reveal a structural parallel with variational formulas in complex pluripotential theory. Our approach differs fundamentally from previous treatments based on Harder--Narasimhan filtrations or arithmetic Okounkov bodies. It places the arithmetic volume at the center of the theory and suggests that asymptotic volume invariants, rather than heights, provide the natural analytic framework for problems in equidistribution and arithmetic dynamics.
