Second-order estimates for collapsed limits of Ricci-flat K\"ahler metrics

TL;DR

This paper establishes a connection between the singularities of Ricci-flat K"ahler metrics in collapsed limits and the holomorphic sectional curvature of the underlying geometry, providing an alternative proof with explicit constants.

Contribution

It offers a new proof linking singularities of Ricci-flat K"ahler metrics to holomorphic sectional curvature, with explicit constants in the estimates.

Findings

Singularities relate to holomorphic sectional curvature.

Provides an alternative proof to existing second-order estimates.

Explicit constants in the estimates are derived.

Abstract

We show that the singularities of the twisted K\"ahler--Einstein metric arising as the long-time solution of the K\"ahler--Ricci flow or in the collapsed limit of Ricci-flat K\"ahler metrics is intimately related to the holomorphic sectional curvature of the reference conical geometry. This provides an alternative proof of the second-order estimate obtained by Gross--Tosatti--Zhang with explicit constants appearing in the divisorial pole.