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.
Categories: math.DG math.CV