Related papers: Stability thresholds for big classes

Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as another invariant, the stability threshold. The latter…

We give a new proof of the fact that the condition of a Fano manifold admitting a K\"ahler-Einstein metric is Zariski-open (provided that the automorphism group is discrete). This proof does not use the characterisation involving stability.…

Let $X$ be a Fano manifold of dimension at least $2$ and $D$ be a smooth divisor in a multiple of the anticanonical class, $\frac1\alpha(-K_X)$ with $\alpha>1$. It is well-known that K\"ahler-Einstein metrics on $X$ with conic singularities…

We annnounce a proof of the fact that a K-stable Fano manifold admits a Kahler-Einstein metric and give a brief outline of the proof.

We consider the problem of existence of constant scalar curvature Kaehler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight…

We give a criterion for the existence of a K\"ahler-Einstein metric on a Fano manifold $M$ in terms of the higher algebraic alpha-invariants $\alpha_{m,k}(M)$.

We give a criterion for the coercivity of the Mabuchi functional for general K\"ahler classes on Fano manifolds in terms of Tian's alpha invariant. This generalises a result of Tian in the anti-canonical case implying the existence of a…

We prove the existence of Kahler-Einstein metric on a K-stable Fano manifold using the recent compactness result on Kahler-Ricci flows. The key ingredient is an algebro-geometric description of the asymptotic behavior of Kahler-Ricci flow…

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…

The global holomorphic \alpha-invariant introduced by Tian is closely related with the study in the existence of Kahler-Einstein metric. We apply the result of Tian, Lu and Zelditch on polarized Kahler metrics to approximate…

We define a new notion of "b-stability" for a polarised algebraic variety, adapted to the existence problem for Kahler-Einstein metrics on Fano manifolds.

An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis…

We relate the global log canonical threshold of a variety with torus action to the global log canonical threshold of its quotient. We apply this to certain Fano varieties and use Tian's criterion to prove the existence of Kahler-Einstein…

We prove the following result: if a $\mathbb{Q}$-Fano variety is uniformly K-stable, then it admits a K\"{a}hler-Einstein metric. We achieve this by modifying Berman-Boucksom-Jonsson's strategy with appropriate perturbative arguments and…

We extend the framework of K-stability (Tian, Donaldson) to more general algebro-geometric setting, such as partial desingularisations of (fixed) singularities, (not necessarily flat) families over higher dimensional base and the classical…

We show that a polarized affine variety admits a Ricci flat K\"ahler cone metric, if and only if it is K-stable. This generalizes Chen-Donaldson-Sun's solution of the Yau-Tian-Donaldson conjecture to K\"ahler cones, or equivalently,…

We show that uniform K-stability is a Zariski open condition in Q-Gorenstein families of Q-Fano varieties. To prove this result, we consider the behavior of the stability threshold in families. The stability threshold (also known as the…

In this note we revisit and extend few classical and recent results on the definition and use of the Futaki invariant in connection with the existence problem for Kaehler constant scalar curvature metrics on polarized algebraic manifolds,…

A reverse H\"older inequality is established on the space of K\"ahler metrics in the first Chern class of a Fano manifold X endowed with Darvas L^{p}-Finsler metrics. The inequality holds under a uniform bound on a twisted Ricci potential…

Let $X$ be any $\mathbb{Q}$-Fano variety and $\mathrm{Aut}(X)_0$ be the identity component of the automorphism group of $X$. Let $\mathbb{G}$ be a connected reductive subgroup of $\mathrm{Aut}(X)_0$ that contains a maximal torus of…