arXiv research

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $Θ$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove…

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

Sharp bounds on Fano varieties' heights proven for specific cases.

Sharp bounds on K-semistable Fano varieties for low dimensions.

In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano Kähler-Einstein manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski open condition and establish the uniqueness for the Gromov-Hausdorff limit …

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 variety of dimension $n$ is at least $\frac{1}{n+1}$ (under mild assumptions) an…

We show that $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.

In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also prove that any special test configuration arises from a log canonical place of a bounded complement and establish properties of any minimizer o…

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.

This is a continuation to the paper [arXiv:1511.08164] in which a problem of minimizing normalized volumes over $\mathbb{Q}$-Gorenstein klt singularities was proposed. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler-Einstein metrics on Fano varieties. In particul…

Characterizes K-semistability for log Fano cone singularities.

Study shows volume limit for K-semistable Fano manifolds.

We prove a version of Jonsson-Mustaţǎ'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 minimizer of the normalized volume function is always quasi-monomial. Applying our techni…

We show that the anti-canonical volume of an $n$-dimensional Kähler-Einstein $\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely $\mathrm{lct}^n\cdot\mathrm{mult}$ for ideals and the normalized volume function for real valuations. This refines a recent result by Fuji…

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 delta-invariant) is a recently introduced invariant that is known to detect the K-s…

Unique K-polystable degenerations for Fano varieties confirmed.

The abstract discusses connections between K-stability, heights, and rational points on Fano varieties.

In this note, using the recent compactness results of Tian and Chen-Donaldson-Sun, we prove the K-semistable version of Yau-Tian-Donaldson correspondence for Fano manifolds.

Study confirms boundedness of certain singularities in log Fano geometry.

We prove two new results on the K-polystability of Q-Fano varieties based on purely algebro-geometric arguments. The first one says that any K-semistable log Fano cone has a special degeneration to a uniquely determined K-polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to…

Finite group action on K-stability results in standard stability.

Proves boundedness of log Fano cone singularities with bounded local volumes.

Equivalence proven between algebraic stability and geometric stability.

New examples of Kähler-Ricci solitons on Fano threefolds with non-trivial moduli found.

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 that if a $\mathbb{Q}$-Fano variety $V$ specially degenerates to a Kähler-Einstein $\mathbb{Q}$-Fano variety $V$, then for any ample Cartier divisor $H=-r^{-1} K_V$ with $r\in \mathbb{Q}_{>0}$, the normalized volume $\widehat{\rm vol}(v)=A_{\mathcal{C}}^n(v)\cdot {\rm vol}(v)$ is globally minimized at the cano…

We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt blowups in order to test uniform K-stability of log Fano pairs. We also discuss the uniform K-stability of two log Fano pairs under crepant…

Proves unique degeneration of log Fano fibration germs.

We prove that a pair (X, D) with X Fano and D a smooth anti-canonical divisor is K-unstable for negative angles, and K-semistable for zero angle.

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 a degeneration to a K-semistable log Fano cone singularity. Moreover, such a mi…

We show that in any $\mathbb{Q}$-Gorenstein flat family of klt singularities, normalized volumes are lower semicontinuous with respect to the Zariski topology. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistabili…

We show that in any $\mathbb{Q}$-Gorenstein flat family of klt singularities, normalized volumes can only jump down at countably many subvarieties. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistability developed…

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…

Let $S$ be a smooth projective variety and $Δ$ a simple normal crossing $\mathbb{Q}$-divisor with coefficients in $(0,1]$. For any ample $\mathbb{Q}$-line bundle $L$ over $S$, we denote by $\mathscr{E}(L)$ the extension sheaf of the orbifold tangent sheaf $T_S(-\log(Δ))$ by the structure sheaf $\mathcal{O}_S$ with the …

K-polystability of a polarised variety is an algebro-geometric notion conjecturally equivalent to the existence of a constant scalar curvature Kähler metric. When a variety is K-unstable, it is expected to admit a "most destabilising" degeneration. In this note we show that if such a degeneration exists, then the limit…

Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.

Study of K-moduli of prime Fano threefolds of genus twelve, proving boundary purely divisorial.

Proves properness of K-moduli spaces for Fano varieties.

The study classifies complex smooth Fano varieties with large pseudoindex.

We examine various examples of horosymmetric manifolds which exhibit interesting properties with respect to canonical metrics. In particular, we determine when the blow-up of a quadric along a linear subquadric admits Kähler-Einstein metrics, providing infinitely many examples of manifolds with no Kähler-Ricci solitons…

We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with [Jia17,BL18], the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension a…

We prove the Yau-Tian-Donaldson's conjecture for any $\mathbb{Q}$-Fano variety that has a log smooth resolution of singularities such that the discrepancies of all exceptional divisors are non-positive. In other words, if such a Fano variety is K-polystable, then it admits a Kähler-Einstein metric. This extends the pre…

New approach proves K-stability of Fano varieties.

The study classifies Fano varieties with specific pseudoindex.

Classifies Fano varieties with large pseudoindex and non-free rational curves.

The study finds Kähler-Einstein metrics on certain Fano varieties of type AIII.

Kähler-Einstein metrics found on special types of symmetric varieties.