Related papers: Stable degenerations of singularities

We prove that any singular K\"ahler--Ricci shrinker $X$ arising as a noncollapsed limit of K\"ahler--Ricci flows admits a natural structure of a locally algebraic complex-analytic variety with log terminal singularities. We then derive…

We prove an algebraic decomposition theorem for the unit group $\mathrm{GL}(R)$ of an arbitrary non-discrete irreducible, continuous ring $R$ (in von Neumann's sense), which entails that every element of $\mathrm{GL}(R)$ is both a product…

In this paper we study the stability of the unique continuation in the case of the wave equation with variable coefficients independent of time. We prove a logarithmic estimate in a arbitrary domain of ${\mathbb R}^{n+1}$, where all the…

We show the properness of the moduli stack of stable surfaces over $\mathbb{Z}[1/30]$, assuming the locally-stable reduction conjecture for stable surfaces. This relies on a local Kawamata--Viehweg vanishing theorem for for 3-dimensional…

We consider matrices with entries in a local ring, Mat(m,n,R). Fix a group action, G on Mat(m,n,R), and a subset of allowed deformations, \Sigma\subseteq Mat(m,n,R). The standard question of Singularity Theory is the…

This is the fourth paper in a series. We prove a conjecture made independently by Boston et al and Shalev. The conjecture asserts that there is an absolute positive constant delta such that if G is a finite simple group acting transitively…

We complement the recent theory of general singular integrals $T$ invariant under the Zygmund dilations $(x_1, x_2, x_3) \mapsto (s x_1, tx_2, st x_3)$ by proving necessary and sufficient conditions for the boundedness and compactness of…

In this paper we extend the notion of Futaki invariant to big and nef classes in such a way that it defines a continuous function on the \K\ cone up to the boundary. We apply this concept to prove that reduced normal crossing singularities…

We discover a "no semistability at infinity" phenomenon for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in the 2-step degeneration theory developed by…

This expository article is based on the author's talk at the Kinosaki Algebraic Geometry Symposium 2025. We discuss some recent progress surrounding stable degeneration in algebraic K-stability theory.

This paper investigates the finite generation of cluster automorphism groups. By applying the pseudo $\mathbb{N}$-grading introduced in our previous work, we establish a sufficient condition for a cluster automorphism group to be finitely…

A half a century ago, George Bergman introduced stunning machinery which would realise any commutative conical monoid as the non-stable $K$-theory of a ring. The ring constructed is ``minimal" or ``universal". Given the success of graded…

We prove that the degree of the CM line bundle for a normal family over a curve with fixed general fibers is strictly minimized if the special fiber is either a smooth projective manifold with a unique cscK metric or ``specially K-stable",…

Tilting objects play a key role in the study of triangulated categories. A famous result due to Iyama and Takahashi asserts that the stable categories of graded maximal Cohen-Macaulay modules over quotient singularities have tilting…

Let $(X, L)$ be a polarized Calabi Yau variety (or canonical polarized variety) with crepant singularity. Suppose $\omega_{KE} \in c_1(L)$ (or $\omega_{KE} \in c_1(K_X)$) is the unique Ricci flat current (or Kahler Einstein current with…

Thurston's ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We…

The article continues the study of the genus of regular languages that the authors introduced in a 2012 paper. Generalizing a previous result, we produce a new family of regular languages on a two-letter alphabet having arbitrary high…

We prove that the log canonical ring of a klt pair of dimension $3$ with $\mathbb{Q}$-boundary over an algebraically closed field of characteristic $p>5$ is finitely generated. In the process we prove log abundance for such pairs in the…

The goal of this paper is to give a general theory of logarithmic Gromov-Witten invariants. This gives a vast generalization of the theory of relative Gromov-Witten invariants introduced by Li-Ruan, Ionel-Parker, and Jun Li, and completes a…

Recent discoveries make it possible to compute the K-theory of certain rings from their cyclic homology and certain versions of their cdh-cohomology. We extend the work of G. Corti\~nas et al. who calculated the K-theory of, in addition to…