Stability theory over toroidal or Novikov type base and Canonical modifications

TL;DR

This paper generalizes stability theories in algebraic geometry to families over toric varieties and Novikov rings, unifying degenerations and extending reduction theorems with applications to moduli of Calabi-Yau cones and Kähler-Ricci solitons.

Contribution

It introduces a unified framework for stability over toric and Novikov base varieties, extending existing theorems and providing new tools for moduli problems.

Findings

Generalized stability theories to toric and Novikov settings.

Extended (semi)stable reduction theorems in higher rank.

Established properness results for moduli of Calabi-Yau cones and Kähler-Ricci solitons.

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 $\Theta$-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.