AI 中文总结
研究卡拉比-丘流形上数值里奇平坦度量缺乏可解释性的问题，提出基于摩尔-彭罗斯伪逆的唐纳森算法新变体，应用于多族流形，发现度量参数在大复结构极限附近的幂律，与SYZ猜想相关。
AI 中文摘要
卡拉比-丘流形上的数值里奇平坦度量越来越精确，但往往缺乏提取理论见解所需的可解释性。本文引入了一种基于摩尔-彭罗斯伪逆的唐纳森算法新变体，作用于环境空间的整体截面而非流形本身。此方法能利用典范单项式基计算可解释的平衡度量。将该算法应用于多个族，发现度量参数在大复结构极限附近遵循新的幂律，并与SYZ猜想预测的格罗莫夫-豪斯多夫度量坍缩相联系。
英文摘要
Numerical Ricci-flat metrics on Calabi-Yau manifolds are becoming increasingly accurate. However, they often lack the interpretability required to extract theoretical insights. In this paper, we introduce a novel variant of Donaldson's algorithm based on the Moore-Penrose pseudo-inverse that operates on the global sections of the ambient space rather than the manifold itself. This approach allows us to use the canonical monomial basis to compute interpretable balanced metrics even at large degrees $k$. Applying our ambient algorithm to multiple families, including the Dwork family and complete intersection Calabi-Yau manifolds, we discover that the metric parameters obey novel power laws near the Large Complex Structure Limit (LCSL). We connect these to the Gromov-Hausdorff metric collapse predicted by the SYZ conjecture.
Comments60 pages, 16 figures
