Balanced Metrics Know About SYZ

Abstract

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.