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On asymptotic behavior of the second Chern forms on degenerating Kähler-Einstein surfaces

Itsuki Tazoe

TL;DR

The work analyzes the asymptotics of second Chern forms along degenerations of Kähler-Einstein surfaces with ADE singularities by examining fiber integrals against test functions. It combines BR’s a priori estimates for cscK metrics, Kronheimer’s ALE gravitational instantons, and Bott–Chern theory to establish Hölder continuity of the fiber integral function F(t) with exponent at least $1/d$ under non-degenerate smoothings. For polarized K3 surfaces, it extends these Hölder bounds to the full disc via an almost Ricci-flat gluing construction and RZ-type degeneration results, revealing quantitative regularity in both cscK and Ricci-flat settings. The results clarify how curvature and Chern-form data behave in degenerations, with implications for moduli of KE surfaces and Calabi–Yau degenerations. The methods integrate hyperkähler geometry, ALE models, and complex-analytic deformation theory to yield precise current-level regularity results for $c_2$ across degenerations.

Abstract

We study an asymptotic behavior of the second Chern forms of canonical metrics on a degenerating family of Kähler 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 Hölder exponent of the function at the origin. Our main results consists of two cases: one is a bound of Hölder exponent along a line for cscK-metrics, using Biquard-Rollin's a priori estimates for cscK-metrics, and the other is a bound of Hölder exponent at the origin for Ricci-flat metrics.

On asymptotic behavior of the second Chern forms on degenerating Kähler-Einstein surfaces

TL;DR

The work analyzes the asymptotics of second Chern forms along degenerations of Kähler-Einstein surfaces with ADE singularities by examining fiber integrals against test functions. It combines BR’s a priori estimates for cscK metrics, Kronheimer’s ALE gravitational instantons, and Bott–Chern theory to establish Hölder continuity of the fiber integral function F(t) with exponent at least under non-degenerate smoothings. For polarized K3 surfaces, it extends these Hölder bounds to the full disc via an almost Ricci-flat gluing construction and RZ-type degeneration results, revealing quantitative regularity in both cscK and Ricci-flat settings. The results clarify how curvature and Chern-form data behave in degenerations, with implications for moduli of KE surfaces and Calabi–Yau degenerations. The methods integrate hyperkähler geometry, ALE models, and complex-analytic deformation theory to yield precise current-level regularity results for across degenerations.

Abstract

We study an asymptotic behavior of the second Chern forms of canonical metrics on a degenerating family of Kähler 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 Hölder exponent of the function at the origin. Our main results consists of two cases: one is a bound of Hölder exponent along a line for cscK-metrics, using Biquard-Rollin's a priori estimates for cscK-metrics, and the other is a bound of Hölder exponent at the origin for Ricci-flat metrics.
Paper Structure (7 sections, 39 theorems, 185 equations)

This paper contains 7 sections, 39 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. ALE Gravitational Instanton
  4. Non-collapsing limits of Kähler-Einstein surfaces
  5. Biquard-Rollin's a priori estimate
  6. Main results
  7. Polarized K3 surface case

Key Result

Theorem 1

Let $\mathcal{X}$ be an irreducible and reduced analytic space of dimension $n+1$. Let $f : \mathcal{X} \to \Delta$ be a holomorphic surjection onto the unit disc $\Delta \subset \mathbb{C}$ and let $X_t := f^{-1}(t) \subset \mathcal{X}$ be the fibre on $t \in \Delta$. Then for every compact subset admits an asymptotic expansion as $t \to 0$, where $T^{r,j}_{m,n}$ are $(1,1)$-currents on $\mathc

Theorems & Definitions (68)

  • Theorem 1: Théorème 1 in Bar
  • Theorem : Theorem \ref{['thn:main']} in section \ref{['sec:Main']}
  • Theorem : Theorem \ref{['thm:main2']} in section \ref{['sec:K3']}
  • Definition 1
  • Theorem 2: Theorem 1.1, Theorem 1.2, Theorem 1.3 in Kr1Kr2
  • Theorem 3: Kr1 Section 3 and Section 4, namely from Corollary 3.12.
  • Lemma 1
  • proof
  • Definition 2
  • Theorem 4: An, Na for (i), BKN for (ii) and Ba for (iii)
  • ...and 58 more