On asymptotic behavior of the second Chern forms on degenerating K\"ahler-Einstein surfaces

TL;DR

This paper investigates the asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces, providing bounds on the H"older exponent in different metric cases with singularities.

Contribution

It offers new bounds on the H"older exponent of fiber integral functions related to second Chern forms on degenerating Kähler surfaces with ADE-singularities.

Findings

Bound of H"older exponent along a line for cscK-metrics

Bound of H"older exponent at the origin for Ricci-flat metrics

Analysis of fiber integral functions on degenerating families

Abstract

We study an asymptotic behavior of the second Chern forms of canonical metrics on a degenerating family of K\"ahler surfaces with the central fibre having ADE-singularities. We investigate a function on the unit disc defined by fiber integrals of the forms with a smooth test function on the family. We show a lower bound of the H\"older exponent of the function at the origin. Our main results consists of two cases: one is a bound of H\"older exponent along a line for cscK-metrics, using Biquard-Rollin's a priori estimates for cscK-metrics, and the other is a bound of H\"older exponent at the origin for Ricci-flat metrics.