arXiv research

Unique K-polystable degenerations for Q-Fano varieties proven.

Classifies K-stable Fano varieties and finds new examples.

Proves CM line bundle positivity for K-stable klt Fano varieties.

Fano varieties get Kähler-Einstein metrics if they are uniformly K-stable.

New proof for K-stability of certain Fano varieties.

We exhibit the first non-trivial concrete examples of Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds in all complex dimensions bigger than two (Fano K-moduli spaces). We also discuss potential applications to explicit study of moduli spaces of K-stable Fano manifolds with large an…

Existence of Kähler-Ricci solitons on smoothable Q-Fano varieties proven.

Uniform K-stability implies Kähler-Einstein metric for Q-Fano varieties.

The notion of Berman-Gibbs stability was originally introduced by Robert Berman for $\mathbb{Q}$-Fano varieties $X$. We show that the pair $(X, -K_X)$ is K-stable (resp. K-semistable) provided that $X$ is Berman-Gibbs stable (resp. semistable).

We show that certain Galois covers of K-semistable Fano varieties are K-stable. We use this to give some new examples of Fano manifolds admitting Kähler-Einstein metrics, including hypersurfaces, double solids and threefolds.

Sharpness of Tian's K-stability criterion demonstrated through specific examples.

We provide a sufficient condition for polarisations of Fano varieties to be K-stable in terms of Tian's alpha invariant, which uses the log canonical threshold to measure singularities of divisors in the linear system associated to the polarisation. This generalises a result of Odaka-Sano in the anti-canonically polari…

The paper proves a CM line bundle is ample on K-moduli spaces.

We give a purely algebro-geometric proof that if the alpha-invariant of a Q-Fano variety X is greater than dim X/(dim X+1), then (X,O(-K_X)) is K-stable. The key of our proof is a relation among the Seshadri constants, the alpha-invariant and K-stability. It also gives applications concerning the automorphism group.

Researchers classify K-stability of log Fano hyperplane arrangements.

Alternative proof of semipositivity and nefness for K-semistable log-Fano pairs.

Proves K-stability and superrigidity of certain singular Fano hypersurfaces.

Proves finitely generated associated graded rings for valuations on log Fano pairs.

The study proves Fano complete intersections' rigidity and stability under certain conditions.

We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in R^n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P. Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donalds…

Found a stable 3D shape with specific properties.

The moduli continuity method is applied to describe K-polystable log Fano pairs.

K-stability proven for a specific type of Fano threefold.

New proof for Fano manifolds, showing rigidity and stability.

Given a one parameter flat family of polarized algebraic varieties, we show that any K-stable limit is unique. In particular, moduli spaces of K-stable polarized varieties are automatically Hausdorff when they exist. We also give a characterization of K-stable limits in terms of the CM line bundle, and some application…

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

Uniformly K-stable toric varieties are asymptotically Chow stable if their Futaki-Ono invariant vanishes.

Solves Yau-Tian-Donaldson conjecture for smooth projective varieties.

Shows CM line bundles are ample on K-stable varieties.

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 generalizes a notion by Fujita and Odaka, and can be used to characterize when a Q-Fa…

Paper confirms Hamilton-Tian conjecture for specific Sasakian manifolds.

We show that if a Fano manifold $M$ is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then $M$ admits a Kähler-Einstein metric. This is a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-Donaldson-Sun, and can be used to ob…

Uniform K-stability holds for close polarizations if original is canonical or anti-canonical.

This is the third and final paper in a series which establish results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2π. We also put all our technical results together to complete the proof of the…

Researchers create a projective space for quasimaps and study its connections to Calabi-Yau fibrations.

The global log canonical threshold (or Tian's alpha-invariant) plays an important role in the geometry of Fano varieties. Tian showed that Fano manifolds with big alpha-invariant can be equipped with a Kahler-Einstein metric. In recent years Donaldson drafted a programme to determine when a smooth Fano variety X admits…

Study links K-stability of certain surfaces to binary forms, proving stability and non-stability conditions.

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 on Fano manifolds. This is in turn based on a general finite dimensional discussi…

Extremal metrics exist if uniformly $K$-stable over models.

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 examples of Fano threefolds admitting a Kahler-Ricci soliton.

Proves Yau-Tian-Donaldson conjecture for certain singular Fano varieties.

Proves properness of K-moduli spaces for Fano varieties.

The study classifies complex smooth Fano varieties with large pseudoindex.

The paper studies Sasaki-Ricci solitons on Sasakian manifolds up to seven dimensions.

New approach proves K-stability of Fano varieties.

The study classifies Fano varieties with specific pseudoindex.