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Stability theory over toroidal or Novikov type base and Canonical modifications

Yuji Odaka

TL;DR

This work develops a comprehensive framework for stability and moduli in multi-parameter, possibly irrational, degenerations by introducing generalized test configurations over toroidal bases, Novikov-type valuation spectra, and higher Θ-stratification. It proves higher-rank semistable reduction theorems, including algebraic and analytic forms, which guarantee canonical modifications of degenerating families to avoid unstable loci and, iteratively, achieve semistability along stratifications. The results yield properness and compactness statements for moduli of Calabi–Yau cones and (singular) Kähler–Ricci solitons, reducing these questions to finite-generation/birational-geometry problems. The framework also connects to birational geometry and metric degeneration, enabling applications to bubbling Kähler metrics and canonical degenerations, and provides a path toward unifying several instability phenomena under a common canonical paradigm.

Abstract

We set up a generalization of ubiquitous one-parameter families in algebraic geometry and their use for stability theories ([GIT, HL, AHLH]) to families over toric varieties and their analytic analogues. The language allows us to reformulate degenerations of ``irrational" direction in the literature as canonical objects in a unified manner. Accordingly, we generalize the (semi)stable reduction-type theorem for $Θ$-stratification in [AHLH] of Langton type to our higher rank setup. We also establish complex analytic analogue of the results. As an infinitesimal analogue of toric spectrum, we also use Novikov type rings as it gives more canonicity but its use can be avoided logically for readers for readers who prefer not to use such rings. As applications, we establish the properness part of the moduli of Calabi-Yau cones (cf., [Od24a]), and also reduce the properness of the moduli of Kahler-Ricci solitons, again to a finite generation type problem in birational geometry.

Stability theory over toroidal or Novikov type base and Canonical modifications

TL;DR

This work develops a comprehensive framework for stability and moduli in multi-parameter, possibly irrational, degenerations by introducing generalized test configurations over toroidal bases, Novikov-type valuation spectra, and higher Θ-stratification. It proves higher-rank semistable reduction theorems, including algebraic and analytic forms, which guarantee canonical modifications of degenerating families to avoid unstable loci and, iteratively, achieve semistability along stratifications. The results yield properness and compactness statements for moduli of Calabi–Yau cones and (singular) Kähler–Ricci solitons, reducing these questions to finite-generation/birational-geometry problems. The framework also connects to birational geometry and metric degeneration, enabling applications to bubbling Kähler metrics and canonical degenerations, and provides a path toward unifying several instability phenomena under a common canonical paradigm.

Abstract

We set up a generalization of ubiquitous one-parameter families in algebraic geometry and their use for stability theories ([GIT, HL, AHLH]) to families over toric varieties and their analytic analogues. The language allows us to reformulate degenerations of ``irrational" direction in the literature as canonical objects in a unified manner. Accordingly, we generalize the (semi)stable reduction-type theorem for -stratification in [AHLH] of Langton type to our higher rank setup. We also establish complex analytic analogue of the results. As an infinitesimal analogue of toric spectrum, we also use Novikov type rings as it gives more canonicity but its use can be avoided logically for readers for readers who prefer not to use such rings. As applications, we establish the properness part of the moduli of Calabi-Yau cones (cf., [Od24a]), and also reduce the properness of the moduli of Kahler-Ricci solitons, again to a finite generation type problem in birational geometry.
Paper Structure (12 sections, 18 theorems, 31 equations)

This paper contains 12 sections, 18 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Generalities on multi-parameter families
  3. Setup of the bases
  4. Generalized test configurations and examples
  5. Higher rank semistable reduction theorem
  6. Toroidal stack-theoretic setup
  7. Higher $\Theta$-semistable reduction theorem
  8. Semistable reduction along higher $\Theta$-stratification: by repetition of Corollary \ref{['gAHLHc']}
  9. Applications
  10. Properness of moduli of Calabi-Yau cones
  11. Properness of moduli of (singular) Kähler-Ricci solitons
  12. Bubbling Kähler metrics

Key Result

Theorem 1.1

Consider a quotient stack $\mathcal{M}$ over a field $\mathbbm{k}$ and its higher $\Theta$-strata $\mathcal{Z}^+\subset \mathcal{M}$ for the cone (see Definition higher.theta.strata). Take any $\mathbbm{k}$-morphism $f\colon \Delta=\mathop{\mathrm{Spec}}\nolimits \mathbbm{k}[[t]]\to \mathcal{M}$ suc which still extends $f^o$ but now sends the closed point $c_{R'}$ to a point outside$Z^+$.

Theorems & Definitions (59)

  • Theorem 1.1: Higher $\Theta$-semistable reduction: cf., Corollary \ref{['gAHLHc']}
  • Definition 2.1: Novikov-type rings and spectra
  • Proposition 2.2
  • proof
  • Definition 2.3: Novikov-type spectra over $R$
  • Definition 2.4: Analytic analogue
  • Lemma 2.5
  • Definition 2.6
  • Example 2.7
  • Example 2.8
  • ...and 49 more