Related papers: A note on the definition of K-stability

We consider a version of Hermitian-Einstein equation but perturbed by a Higgs field with a solution called a Donaldson-Thomas instanton on compact K\"ahler threefolds. The equation could be thought of as a generalization of the Hitchin…

We algebraically prove K-stability of polarized Calabi-Yau varieties and canonically polarized varieties with mild singularities. In particular, the} "stable varieties" introduced by Kollar-Shepherd-Barron and Alexeev, which form compact…

In this paper, we prove that any polarized K-stable manifold is CM-stable. This extends what I did for Fano manifolds in my 2012 paper.

We formulate a notion of K-stability for K\"ahler manifolds, and prove one direction of the Yau-Tian-Donaldson conjecture in this setting. More precisely, we prove that the Mabuchi functional being bounded below (resp. coercive) implies…

We study the existence of extremal K\"ahler metrics on K\"ahler manifolds. After introducing a notion of relative K-stability for K\"ahler manifolds, we prove that K\"ahler manifolds admitting extremal K\"ahler metrics are relatively…

In this paper we study K-polystability of arbitrary (possibly non-projective) compact K\"ahler manifolds admitting holomorphic vector fields. As a main result, we show that existence of a constant scalar curvature K\"ahler (cscK) metric…

Let ${\rm k}$ be an algebraically closed field of characteristic 0 and $G$ a connect, reductive group over it. Let $X$ be a projective $G$-variety of complexity 1. We classify $G$-equivariant normal test configurations of $X$ with integral…

For any flat projective family $(\mX,\mL)\rightarrow C$ such that the generic fibre $\mX_\eta$ is a klt Q-Fano variety and $\mL|_{\mX_\eta}\sim_{Q}-K_{X_{\eta}}$, we use the techniques from the minimal model program (MMP) to modify the…

We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples,…

We show that any $n$-dimensional Fano manifold $X$ with $\alpha(X)=n/(n+1)$ and $n\geq 2$ is K-stable, where $\alpha(X)$ is the alpha invariant of $X$ introduced by Tian. In particular, any such $X$ admits K\"ahler-Einstein metrics and the…

We survey some recent developments in the direction of the Yau-Tian-Donaldson conjecture, which relates the existence of constant scalar curvature K\"ahler metrics to the algebro-geometric notion of K-stability. The emphasis is put on the…

We use the equivariant localization formula to prove that the Donaldson-Futaki invariant of a compact smooth (K{\"a}hler) test configuration coincides with the Futaki invariant of the induced action on the central fiber when this fiber is…

We study the question for which commutative ring spectra $A$ the tensor of a simplicial set $X$ with $A$, $X \otimes A$, is a stable invariant in the sense that it depends only on the homotopy type of $\Sigma X$. We prove several structural…

Let $G$ be a connected, complex reductive Lie group and $X$ a $\mathbb Q$-Fano $G$-spherical variety. In this paper we compute the weighed non-Archimedean functionals of a $G$-equivariant normal test configurations of $X$ via combinatory…

Using a quasi-linear version of Hodge theory, holomorphic vector bundles in a neighbourhood of a given polystable bundle on a compact Kaehler manifold are shown to be (poly)stable if and only if their corresponding classes are (poly)stable…

In this work, kinks with non-canonical kinetic energy terms are studied in a type of two-dimensional dilaton gravity model. The linear stability issue is generally discussed for arbitrary static solutions, and the stability criteria are…

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…

A linear dynamical system is called $k$-positive if its dynamics maps the set of vectors with up to $k-1$ sign variations to itself. For $k=1$, this reduces to the important class of positive linear systems. Since stable positive linear…

We introduce a theory of uniform K-stability for big line bundles on smooth projective varieties. This extends the existing theory both for varieties with ample line bundles, and for varieties with big anticanonical class. Our main result…

We show that a polarised manifold with a constant scalar curvature K\"ahler metric and discrete automorphisms is K-stable. This refines the K-semistability proved by S. K. Donaldson.