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Families of singular Chern-Ricci flat metrics

Chung-Ming Pan

TL;DR

The paper extends pluripotential theory for complex Monge–Ampère equations to families of hermitian, possibly singular spaces, and proves uniform a priori estimates for Chern–Ricci flat metrics. It develops local and global L^∞ bounds, uniform Skoda integrability, and a robust volume-capacity framework that persists under degeneration, under a sup–L^1 type assumption and canonical-singularity hypotheses. These tools yield uniform control of MA potentials and normalization constants across conifold-type transitions and Calabi–Yau families, enabling the construction and comparison of Chern–Ricci flat metrics in degenerate settings. The results illuminate the behavior of singular Calabi–Yau metrics in moduli and smoothing families, with implications for non-Kähler Calabi–Yau geometry and conifold transitions.

Abstract

We prove uniform a priori estimates for degenerate complex Monge-Ampère equations on a family of hermitian varieties. This generalizes a theorem of Di Nezza-Guedj-Guenancia to hermitian contexts. The main result can be applied to study the uniform boundedness of Chern-Ricci flat potentials in conifold transitions.

Families of singular Chern-Ricci flat metrics

TL;DR

The paper extends pluripotential theory for complex Monge–Ampère equations to families of hermitian, possibly singular spaces, and proves uniform a priori estimates for Chern–Ricci flat metrics. It develops local and global L^∞ bounds, uniform Skoda integrability, and a robust volume-capacity framework that persists under degeneration, under a sup–L^1 type assumption and canonical-singularity hypotheses. These tools yield uniform control of MA potentials and normalization constants across conifold-type transitions and Calabi–Yau families, enabling the construction and comparison of Chern–Ricci flat metrics in degenerate settings. The results illuminate the behavior of singular Calabi–Yau metrics in moduli and smoothing families, with implications for non-Kähler Calabi–Yau geometry and conifold transitions.

Abstract

We prove uniform a priori estimates for degenerate complex Monge-Ampère equations on a family of hermitian varieties. This generalizes a theorem of Di Nezza-Guedj-Guenancia to hermitian contexts. The main result can be applied to study the uniform boundedness of Chern-Ricci flat potentials in conifold transitions.
Paper Structure (29 sections, 23 theorems, 105 equations)

This paper contains 29 sections, 23 theorems, 105 equations.

Table of Contents

  1. Preliminaries
  2. Smooth forms and currents on singular spaces
  3. Plurisubharmonic functions
  4. Lelong numbers
  5. Monge--Ampère capacities
  6. Uniform $L^\infty$-estimate
  7. Local $L^\infty$-estimate
  8. Global $L^\infty$-estimate
  9. Upper bound of $c$
  10. Domination principle
  11. Subsolution estimate
  12. $L^\infty$-estimate
  13. Proof of Theorem \ref{['bigthm:uniform_a_priori_estimate']}
  14. Uniform Skoda's integrability theorem
  15. Local uniform Skoda's estimate
  16. ...and 14 more sections

Key Result

Theorem A

Let $\pi: (\mathcal{X},\omega) \rightarrow \mathbb{D}$ be a family of compact, locally irreducible, hermitian varieties and $0\leq f_t \in L^p(X_t,\omega_t^n)$ be a family of densities. Assume that $\pi: (\mathcal{X},\omega) \rightarrow \mathbb{D}$ fits into Conjecture conj:SL and $(f_t)_{t \in \mat For each $t \in \mathbb{D}$, let the pair $(\varphi_t,c_t) \in \left(\mathop{\mathrm{PSH}}\nolimits

Theorems & Definitions (45)

  • Theorem A
  • Conjecture (SL)
  • Proposition B
  • Theorem C
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4: Demailly_1982
  • Definition 1.5
  • Theorem 1.6: Bedford_Taylor_1982
  • ...and 35 more