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K-moduli of quasimaps and on quasi-projectivity of moduli of K-stable Calabi-Yau fibrations over curves

Kenta Hashizume, Masafumi Hattori

TL;DR

The paper constructs a projective K-moduli space for log Fano quasimaps to a fixed projective target and relates it to the K-moduli of Calabi–Yau fibrations over curves with fibers such as Abelian or irreducible holomorphic symplectic varieties. It proves that the CY-fibration moduli semnormalization is quasi-projective and that the CM line bundle is ample on the normalization, via a quasi-finite period-mapping bridge to the quasimap moduli. Central to the approach is the notion of uniform adiabatic K-stability for klt–trivial fibrations and the use of quasimaps to absorb degree jumps in moduli maps, enabling good moduli theory in settings where ordinary maps fail. The work also develops Θ-reductivity, S-completeness, and CM-line positivity for the quasimap moduli, and extends K-moduli to arbitrary weights, building a robust framework that connects stability, moduli, and period data across Calabi–Yau fibered families. Overall, it provides a pathway to quasi-projectivity results for nested K-moduli problems and highlights the adiabatic CM-limit mechanism linking fibered Calabi–Yau families with log Fano quasimaps.

Abstract

We construct a projective K-moduli space of quasimaps with a certain log Fano condition, which is regarded as a rational map from $\mathbb{P}^1$ to a projective space. Moreover, we investigate relationships between the K-moduli of quasimaps and the K-moduli of Calabi-Yau fibrations over curves of negative Kodaira dimension constructed by the authors when general fibers are Abelian varieties or irreducible holomorphic symplectic manifolds. As an application, we obtain the entire quasi-projectivity of the seminormalization and the ampleness of the CM line bundle on the normalization of the K-moduli space of Calabi-Yau fibrations in this case.

K-moduli of quasimaps and on quasi-projectivity of moduli of K-stable Calabi-Yau fibrations over curves

TL;DR

The paper constructs a projective K-moduli space for log Fano quasimaps to a fixed projective target and relates it to the K-moduli of Calabi–Yau fibrations over curves with fibers such as Abelian or irreducible holomorphic symplectic varieties. It proves that the CY-fibration moduli semnormalization is quasi-projective and that the CM line bundle is ample on the normalization, via a quasi-finite period-mapping bridge to the quasimap moduli. Central to the approach is the notion of uniform adiabatic K-stability for klt–trivial fibrations and the use of quasimaps to absorb degree jumps in moduli maps, enabling good moduli theory in settings where ordinary maps fail. The work also develops Θ-reductivity, S-completeness, and CM-line positivity for the quasimap moduli, and extends K-moduli to arbitrary weights, building a robust framework that connects stability, moduli, and period data across Calabi–Yau fibered families. Overall, it provides a pathway to quasi-projectivity results for nested K-moduli problems and highlights the adiabatic CM-limit mechanism linking fibered Calabi–Yau families with log Fano quasimaps.

Abstract

We construct a projective K-moduli space of quasimaps with a certain log Fano condition, which is regarded as a rational map from to a projective space. Moreover, we investigate relationships between the K-moduli of quasimaps and the K-moduli of Calabi-Yau fibrations over curves of negative Kodaira dimension constructed by the authors when general fibers are Abelian varieties or irreducible holomorphic symplectic manifolds. As an application, we obtain the entire quasi-projectivity of the seminormalization and the ampleness of the CM line bundle on the normalization of the K-moduli space of Calabi-Yau fibrations in this case.
Paper Structure (27 sections, 64 theorems, 204 equations)

This paper contains 27 sections, 64 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Positivity of CM line bundles and K-moduli conjecture
  3. Moduli of maps and fibered varieties
  4. Main results and key idea: Quasimaps
  5. Related works
  6. Structure of the paper
  7. Preliminaries
  8. Log pairs and klt-trivial fibration
  9. K-stability
  10. Moduli schemes and functors
  11. Artin stacks and good moduli spaces
  12. Moduli stacks and spaces of polarized varieties
  13. CM line bundle
  14. Klt Calabi-Yau varieties
  15. Abelian varieties
  16. ...and 12 more sections

Key Result

Theorem 1.2

Fix a closed immersion $\iota\colon X\hookrightarrow \mathbb{P}^N$. Then a moduli stack $\mathcal{M}^{\mathrm{Kss,qmaps}}_\iota$ of K-semistable log Fano quasimaps from $\mathbb{P}^1$ to $X$ with fixed numerical data is an Artin stack of finite type. Furthermore, $\mathcal{M}^{\mathrm{Kss,qmaps}}_\i

Theorems & Definitions (185)

  • Conjecture 1.1: K-moduli conjecture
  • Theorem 1.2: $=$ Theorem \ref{['thm--K-moduli-of-log-Fano-quasimaps']}
  • Theorem 1.3: cf. Theorems \ref{['thm--quasi--map--construction--moduli--map']}, \ref{['thm--quasi-finiteness--of--two--moduli']} and Corollary \ref{['cor--final']}
  • Remark 1.4
  • Example 1.5
  • Definition 2.1: Singularities of log pairs
  • Definition 2.2: Relative Mumford divisor kollar-moduli
  • Definition 2.3: Log canonical threshold
  • Definition 2.4: Log minimal model, log canonincal model
  • Lemma 2.5
  • ...and 175 more