Related papers: Finite generation for valuations computing stabili…

The purpose of this paper is to clarify all of the uniformly relatively Ding stable toric Fano threefolds and fourfolds as well as unstable ones. The key player in our classification result is the Mabuchi constants, which can be calculated…

Let $(X, \Delta)$ be a log Fano pair with standard coefficients endowed with a singular K\"ahler--Einstein metric. We show that the adapted tangent sheaf $\mathcal{T}_{X, \Delta, f}$ and the adapted canonical extension $\mathcal{E}_{X,…

Given a klt singularity $x\in (X, D)$, we show that a quasi-monomial valuation $v$ with a finitely generated associated graded ring is the minimizer of the normalized volume function $\widehat{\rm vol}_{(X,D),x}$, if and only if $v$ induces…

We study logarithmic K-stability for pairs by extending the formula for Donaldson-Futaki invariants to log setting. We also provide algebro-geometric counterparts of recent results of existence of Kahler-Einstein metrics with cone…

We give a simple necessary and sufficient condition for uniform K-stability of $\mathbb{Q}$-Fano varieties.

We give a classification of Gorenstein Fano bi-equivariant compactifications of semisimple complex Lie groups with rank two, and determine which of them are equivariant K-stable and admit (singular) K\"{a}hler-Einstein metrics. As a…

For Fano varieties, significant progress has been made recently in the study of $K$-stability, while the understanding of the weaker but more algebraic concept of $(-K)$-slope stability remains intricate. For instance, a conjecture…

We introduce a notion of K-stability for adjoint foliated structures via test configurations and the foliated Donaldson-Futaki invariant. We prove reduction to special test configurations for adjoint Fano foliated structures by showing that…

In this article, we completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable or not.

We prove a version of Jonsson-Musta\c{t}\v{a}'s Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm Chi Li's conjecture that a…

In 1987, the $\alpha$-invariant theorem gave a fundamental criterion for existence of Kahler-Einstein metrics on smooth Fano manifolds. In 2012, Odaka-Sano extended the framework to $\mathbb{Q}$-Fano varieties in terms of K-stability, and…

Extending previous results, we prove that for $n \ge 5$ all hypersurfaces of degree $n+1$ in ${\mathbb P}^{n+1}$ with isolated ordinary double points are birational superrigid and K-stable, hence admit a weak K\"ahler--Einstein metric.

We consider coupled K\"ahler-Einstein metrics and weighted solitons on Fano manifolds. These are natural generalizations of K\"ahler-Einstein metrics. As in the case of K\"ahler-Einstein metrics, the existence is known to be equivalent to…

Let X be a Fano manifold. G.Tian proves that if X admits a Kaehler-Einstein metric, then it satisfies two different stability conditions: one involving the Futaki invariant of a special degeneration of X, the other Hilbert-Mumford-stability…

We generalise partial results about the Yau-Tian-Donaldson correspondence on ruled manifolds to bundles whose fibre is a classical flag variety. This is done using Chern class computations involving the combinatorics of Schur functors. The…

This four-pages note is an invitation to explore explicit K-stability for arbitrary K\"ahler classes of low dimension and low rank spherical varieties. We apply our simple combinatorial criterion of K-stability of rank one spherical…

We give a complete criterion for the existence of generalized K\"ahler Einstein metrics on toric Fano manifolds from view points of a uniform stability in a sense of GIT and the properness of a functional on the space of K\"ahler metrics.

We prove the $K$-polystability of all smooth complex Fano threefolds admitting an effective action of $\text{SL}_2$ but not of a 2-torus or 3-torus. In particular, the existence of K\"{a}hler-Einstein metrics on varieties in the families…

We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature K\"ahler metric. This…

We prove that every birationally superrigid Fano variety whose alpha invariant is greater than (resp. no smaller than) $\frac{1}{2}$ is K-stable (resp. K-semistable). We also prove that the alpha invariant of a birationally superrigid Fano…