Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.39996

A Yau-Tian-Donaldson correspondence for degenerations

Gabriel Frey
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove an equivalence between the existence of a unique Kähler-Einstein metric on a degeneration of Fano varieties and uniform CM minimization. This proves a discretely valued analogue of the (uniform) Yau-Tian-Donaldson conjecture, in the sense of non-Archimedean geometry. As a main tool, we prove that Ding stability of a degeneration implies uniform Ding minimization, proving a converse to work of Blum-Xu. We also prove an analogous result in geometric invariant theory (GIT), stating that uniform GIT height minimization of a family of points is equivalent to GIT stability, proving a converse to work of Wang, Wang-Xu and the author.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Gabriel Frey. 2026-10-29. A Yau-Tian-Donaldson correspondence for degenerations. https://arxiv.org/abs/2610.39996

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Wall divisors on irreducible symplectic orbifolds of Nikulin-type

We determine the wall divisors on irreducible symplectic orbifolds which are deformation equivalent to a special type of examples, called Nikulin orbifolds. The Nikulin orbifolds are obtained as partial resolutions in codimension 2 of a quotient by a symplectic involution of a Hilbert scheme of 2 points on a K3 surface. This builds on the previous article arXiv:2009.04873 in which the theory of wall divisors was generalized to orbifold singularities.

math.AG↗
arXiv2026

Prekosmic Grothendieck/Galois Categories

We establish a generalized version of the duality between groups and the categories of their representations on sets. Given an abstract symmetric monoidal category $K$ called Galois prekosmos, we define pre-Galois objects in $K$ and study the categories of their representations internal to $K$. The motivating example of $K$ is the cartesian monoidal category $\textit{Set}$ of sets, and pre-Galois objects in $\textit{Set}$ are groups. We present an axiomatic definition of pre-Galois $K$-categories, which is a complete abstract characterization of the categories of representations of pre-Galois objects in $K$. The category of covering spaces over a well-connected topological space is a prototype of a pre-Galois $\textit{Set}$-category. We establish a perfect correspondence between pre-Galois objects in $K$ and pre-Galois $K$-categories pointed with pre-fiber functors. We also establish a generalized version of the duality between flat affine group schemes and the categories of their linear representations. Given an abstract symmetric monoidal category $K$ called Grothendieck prekosmos, we define what are pre-Grothendieck objects in $K$ and study the categories of their representations internal to $K$. The motivating example of $K$ is the symmetric monoidal category $\textit{Vec}_k$ of vector spaces over a field $k$, and pre-Grothendieck objects in $\textit{Vec}_k$ are affine group $k$-schemes. We present an axiomatic definition of pre-Grothendieck $K$-categories, which is a complete abstract characterization of the categories of representations of pre-Grothendieck objects in $K$. The indization of a neutral Tannakian category over a field $k$ is a prototype of a pre-Grothendieck $\textit{Vec}_k$-category. We establish a perfect correspondence between pre-Grothendieck objects in $K$ and pre-Grothendieck $K$-categories pointed with pre-fiber functors.

math.AG↗