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On positivity of CM line bundles on the moduli space of klt good minimal models with $κ=1$

Masafumi Hattori

TL;DR

This work advances the K-moduli program for klt good minimal models with κ = 1 by proving the CM line bundle becomes ample after normalization and by establishing quasi-projectivity for the seminormalization under mild fiber assumptions. A new moduli space N of numerical equivalence classes is built to overcome non-quasi-finiteness and to connect the moduli of good minimal models with the K-moduli of quasimaps, enabling a CM-line positivity comparison via a commutative diagram. The authors also prove projectivity of moduli spaces of stable quasimaps, extend Viehweg’s numerical moduli to Q-Cartier divisors in the klt setting, and construct BB-type compactifications using the B-semiampleness results of BFMT. A morphism from the moduli of good minimal models to the moduli of quasimaps is established, with CM-line positivity transported along this map, underpinning a concrete path toward resolving the K-moduli conjecture in κ = 1 cases. Overall, the paper integrates moduli-theoretic, stability, and compactification techniques to produce ampleness results for CM line bundles and to unify several moduli frameworks under a common numerical- and stability-driven approach.$κ=1$ and K-stability concepts are central, with meaningful implications for moduli positivity and the global geometry of these spaces.

Abstract

We study the positivity of CM line bundles on the coarse moduli space of Kawamata log terminal (klt) good minimal models with Kodaira dimension one. We prove that the seminormalization of the moduli space is quasi-projective under a mild assumption on the general fibers of good minimal models. Moreover, we show that the CM line bundle becomes ample after normalization. A key new ingredient is the construction of a moduli space of numerical equivalence classes, which is an extension of the work of Viehweg and allows us to bypass the failure of quasi-finiteness in the approach of the previous work by Hashizume and the author. We also establish the projectivity of the moduli space of $ε$-stable quotients, which is introduced by Toda, to a projective space, which plays a central role in our method. This particular situation is encompassed by our general framework of K-moduli of quasimaps.

On positivity of CM line bundles on the moduli space of klt good minimal models with $κ=1$

TL;DR

This work advances the K-moduli program for klt good minimal models with κ = 1 by proving the CM line bundle becomes ample after normalization and by establishing quasi-projectivity for the seminormalization under mild fiber assumptions. A new moduli space N of numerical equivalence classes is built to overcome non-quasi-finiteness and to connect the moduli of good minimal models with the K-moduli of quasimaps, enabling a CM-line positivity comparison via a commutative diagram. The authors also prove projectivity of moduli spaces of stable quasimaps, extend Viehweg’s numerical moduli to Q-Cartier divisors in the klt setting, and construct BB-type compactifications using the B-semiampleness results of BFMT. A morphism from the moduli of good minimal models to the moduli of quasimaps is established, with CM-line positivity transported along this map, underpinning a concrete path toward resolving the K-moduli conjecture in κ = 1 cases. Overall, the paper integrates moduli-theoretic, stability, and compactification techniques to produce ampleness results for CM line bundles and to unify several moduli frameworks under a common numerical- and stability-driven approach. and K-stability concepts are central, with meaningful implications for moduli positivity and the global geometry of these spaces.

Abstract

We study the positivity of CM line bundles on the coarse moduli space of Kawamata log terminal (klt) good minimal models with Kodaira dimension one. We prove that the seminormalization of the moduli space is quasi-projective under a mild assumption on the general fibers of good minimal models. Moreover, we show that the CM line bundle becomes ample after normalization. A key new ingredient is the construction of a moduli space of numerical equivalence classes, which is an extension of the work of Viehweg and allows us to bypass the failure of quasi-finiteness in the approach of the previous work by Hashizume and the author. We also establish the projectivity of the moduli space of -stable quotients, which is introduced by Toda, to a projective space, which plays a central role in our method. This particular situation is encompassed by our general framework of K-moduli of quasimaps.
Paper Structure (22 sections, 61 theorems, 99 equations)

This paper contains 22 sections, 61 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Background of K-moduli conjecture and positivity of the CM line bundle
  3. Main result
  4. Main result in a precise form
  5. Quasimaps
  6. Technical core of this paper
  7. Related works
  8. Structure of the paper
  9. Preliminaries
  10. Birational geometry and K-stability
  11. $\mathbb{Q}$-Gorenstein family
  12. CM line bundle
  13. Representable functors
  14. Moduli stacks and spaces of polarized varieties
  15. K-moduli of klt--trivial fibrations over curves
  16. ...and 7 more sections

Key Result

Theorem 1.2

Let $\mathscr{M}_{\kappa=1}$ be a connected component of the K-moduli space parameterizing polarized klt good minimal models $f\colon (X,A)\to C$ with $\kappa=1$. Take the open subspace $\mathscr{M}\subset \mathscr{M}_{\kappa=1}$ such that $f$ belongs to $\mathscr{M}$ if and only if the generic fibe

Theorems & Definitions (154)

  • Conjecture 1.1: K-moduli conjecture
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5: Proposition \ref{['lem--quasimap--stack']}, Theorem \ref{['thm--qm--cm--positive']}
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 144 more