发表机构
Nguyen Trai University（阮梯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出多势理论框架，通过m-Hessian容量等估计连接复势理论与几何代数有界性，以研究极化Calabi--Yau流形的有限性，并指出关键待发展步骤。
AI 中文摘要
我们提出了一个多势理论框架，用于研究极化Calabi--Yau流形的有限性问题。核心思想是利用全局m-Hessian容量作为复势理论与全局几何有界性之间的定量中间环节。我们构建了一系列均匀估计，将m-Hessian容量、体积-容量不等式、容量-周长估计、Sobolev界以及相关Ricci-flat Kähler度量的非坍缩联系起来。在适当的正则性假设下，这些估计导致均匀的直径和曲率控制，从而得到度量紧性。然后我们描述了论证的代数部分。具有有界次数的均匀射影嵌入，通过Macaulay--Gotzmann理论，导致只有有限多个可能的Hilbert多项式，从而得到有限的Hilbert概形集合。Ehresmann纤维化定理随后将光滑代数族的紧性转化为微分同胚和拓扑类型的有限性。所得到的框架在Calabi--Yau有限性的势理论方法中隔离了主要的解析和代数瓶颈。特别是，容量-周长估计以及从先前的度量估计推导均匀代数嵌入数据构成了需要进一步发展的关键步骤。因此，本文提供了一个结构化的程序，将复Hessian势理论与几何和代数有界性联系起来，而不是假设这些蕴含关系是自动成立的。
英文摘要
We propose a pluripotential-theoretic framework for studying finiteness questions for polarized Calabi--Yau manifolds. The central idea is to use global \(m\)-Hessian capacities as a quantitative intermediate between complex potential theory and global geometric boundedness. We formulate a chain of uniform estimates connecting \(m\)-Hessian capacity, volume--capacity inequalities, capacity--perimeter estimates, Sobolev bounds, and non-collapsing of the associated Ricci-flat Kähler metrics. Under suitable regularity assumptions, these estimates lead to uniform diameter and curvature control and hence to metric compactness. We then describe the algebraic part of the argument. Uniform projective embeddings with bounded degree lead, through the Macaulay--Gotzmann theory, to only finitely many possible Hilbert polynomials and hence to a finite collection of Hilbert schemes. Ehresmann's fibration theorem then converts boundedness of the smooth algebraic families into finiteness of diffeomorphism and topological types. The resulting framework isolates the main analytic and algebraic bottlenecks in a potential-theoretic approach to Calabi--Yau finiteness. In particular, the capacity--perimeter estimate and the derivation of uniform algebraic embedding data from the preceding metric estimates constitute the essential steps requiring further development. Thus the paper provides a structured program linking complex Hessian potential theory with geometric and algebraic boundedness, rather than assuming that these implications are automatic.
