Second-order estimates for collapsed limits of Ricci-flat Kähler metrics

Abstract

We show that the singularities of the twisted Kähler-Einstein metric arising as the longtime solution of the Kähler-Ricci flow or in the collapsed limit of Ricci-flat Kähler metrics is intimately related to the holomorphic sectional curvature of reference conical geometry. This provides an alternative proof of the second-order estimate obtained by Gross-Tosatti-Zhang [14] with explicit constants appearing in the divisorial pole.