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Boundedness of stable minimal models with klt singularities

Minzhe Zhu

TL;DR

This work proves that klt stable minimal models with fixed dimension, Iitaka volume, and coefficients from a DCC set form a bounded family, while also obtaining uniform ε-lc control of their singularities. The authors extend effective adjunction to real coefficients, develop a decomposition technique for ℝ-generalized pairs, and leverage the induced adjunction to bound both the base and fibers in a fibration. A key innovation is controlling the non-lc locus and ensuring nefness in a dimension-by-dimension fashion, enabling the global boundedness result. The results advance the moduli theory for intermediate Kodaira dimension varieties and provide foundational steps toward moduli of stable minimal models with singularities beyond the rational-coefficient setting.

Abstract

We investigate the singularities and boundedness of a special kind of algebraic varieties so-called stable minimal models, which are constructed and studied by Birkar. Given a klt stable minimal model with bounded relative volume, if we fix the dimension, Iitaka volume, and a DCC set controlling coefficients, then we show that the singularities of the klt stable minimal model can be controlled uniformly. Furthermore, we prove that with certain bounded data, stable minimal models with klt singularities form a bounded family.

Boundedness of stable minimal models with klt singularities

TL;DR

This work proves that klt stable minimal models with fixed dimension, Iitaka volume, and coefficients from a DCC set form a bounded family, while also obtaining uniform ε-lc control of their singularities. The authors extend effective adjunction to real coefficients, develop a decomposition technique for ℝ-generalized pairs, and leverage the induced adjunction to bound both the base and fibers in a fibration. A key innovation is controlling the non-lc locus and ensuring nefness in a dimension-by-dimension fashion, enabling the global boundedness result. The results advance the moduli theory for intermediate Kodaira dimension varieties and provide foundational steps toward moduli of stable minimal models with singularities beyond the rational-coefficient setting.

Abstract

We investigate the singularities and boundedness of a special kind of algebraic varieties so-called stable minimal models, which are constructed and studied by Birkar. Given a klt stable minimal model with bounded relative volume, if we fix the dimension, Iitaka volume, and a DCC set controlling coefficients, then we show that the singularities of the klt stable minimal model can be controlled uniformly. Furthermore, we prove that with certain bounded data, stable minimal models with klt singularities form a bounded family.
Paper Structure (20 sections, 14 theorems, 29 equations)

This paper contains 20 sections, 14 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Sketch of proofs
  3. Acknowledgements
  4. Preliminaries
  5. Divisors
  6. (Generalized) Pairs and Singularities
  7. Adjunction formulas for fiber spaces
  8. Bounded families
  9. Families of generalized pairs
  10. Decomposition of $\mathbb{R}$-coefficient generalized pairs in ${\mathcal{F}}_{gklt}(d,\Phi,v)$
  11. Relative nefness
  12. Effective adjunction formula with real coefficients
  13. Cartier index of moduli divisors
  14. Finite coefficients case
  15. DCC coefficients case
  16. ...and 5 more sections

Key Result

Theorem 1.5

Let $d\in \mathbb{N}$, $u,v,w\in \mathbb{R}^{>0}$, and $\Phi\subset \mathbb{R}^{\geq 0}$ be a DCC set. Let $(X,B),A\in {\mathcal{S}}_{klt}(d, \Phi,\leq u,v)$ satisfy that the intersection number $(K_X+B)^i\cdot A^{d-i}\leq w$ for any $0\leq i\leq d$. Then the set of such $(X,B),A$ forms a bounded fa

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1: ACC sets and DCC sets
  • Definition 2.2: Coefficients of divisors
  • Definition 2.3: Contractions
  • ...and 34 more