Related papers: On K-polystability for log del Pezzo pairs of Maed…

In an algebro-geometric way, we completely determine whether smooth del Pezzo surfaces are K-(semi)stable or not.

In this paper, we study compactifications of the moduli of smooth del Pezzo surfaces using K-stability and the line arrangement. We construct K-moduli of log del Pezzo pairs with sum of lines as boundary divisors, and prove that for…

We completely classify K-stability of log del Pezzo hypersurfaces with index 2.

Motivated by the problem for the existence of K\"ahler-Einstein edge metrics, Cheltsov and Rubinstein conjectured the K-polystability of asymptotically log Fano varieties with small cone angles when the anti-log-canonical divisors are not…

We study the K-moduli of log del Pezzo pairs formed by a del Pezzo surface of degree $d$ and an anti-canonical divisor. These moduli spaces naturally depend on one parameter, providing a natural problem in variations of K-moduli spaces. For…

For every integer $a \geq 2$, we relate the K-stability of hypersurfaces in the weighted projective space $\mathbb{P}(1,1,a,a)$ of degree $2a$ with the GIT stability of binary forms of degree $2a$. Moreover, we prove that such a…

For an arbitrary ample divisor A in smooth del Pezzo surface S of degree 1, we verify the condition of the polarization (S,A) to be K-stable and it is a simple numerical condition.

Asymptotically log Fano pairs were introduced by Cheltsov and Rubinstein, generalising a definition of Maeda. They have received attention in the last decade within the theory of K-stability, as they approximate log Calabi Yau pairs while…

In this paper, we develop an algebraic K-stability theory (e.g. special test configuration theory and optimal destabilization theory) for log Fano $\mathbb R$-pairs, and construct a proper K-moduli space to parametrize K-polystable log Fano…

We give a simple sufficient condition for K-stability of polarized del Pezzo surfaces and for the existence of a constant scalar curvature Kahler metric in the Kahler class corresponding to the polarization.

We prove that a log Fano cone $(X,\Delta,\xi_0)$ satisfying $\delta_\mathbb{T}(X,\Delta,\xi_0)\ge 1$ is K-polystable for normal test configurations if and only if it is K-polystable for special test configurations. We also establish the…

There are several variations of the definition of log del Pezzo pairs in the literature. We define their suitable smooth models, and we show that they are the same. In particular, we obtain a characterization of smooth log del Pezzo pairs…

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

We give a classification of all pairs (X,v) of Gorenstein del Pezzo surfaces X and vector fields v which are K-stable in the sense of Berman-Nystrom and therefore are expected to admit a Kahler-Ricci solition. Moreover, we provide some new…

We show that the pair $(X, -K_X)$ is K-unstable for a del Pezzo manifold $X$ of degree five with dimension four or five. This disprove a conjecture of Odaka and Okada.

We prove the existence of singular del Pezzo surfaces that are neither K-semistable nor contain any anticanonical polar cylinder.

We give examples of K-unstable singular del Pezzo surfaces which are weighted hypersurfaces with index 2.

We compute the $\delta$-invariant for pairs $(\mathbb{P}^2, \lambda C_d)$, where $C_d$ is a plane curve of degree $d \leq 4$. These computations provide new examples of $K$-stable and $K$-semistable log Fano pairs, and contribute to the…

We prove that a normal hyperplane section of the Segre variety $\Sigma_{m, n}$ is K-unstable with respect to any polarization if $m\neq n$ or it is not smooth.

This is essentially an expository note based on S. Paul's works on the stability of pairs. Its connection to K-stability will be also discussed.