arXiv research

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

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 algebraic version of Hamilton-Tian conjecture for log Fano pairs.

Study shows polystability of tangent and canonical sheaves on Kähler-Einstein log Fano pairs.

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

Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional assumption of log smoothness, and give a complete classification of two dimension…

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…

The 'moduli continuity method' permits an explicit algebraisation of the Gromov-Hausdorff compactification of Kähler-Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the 'log setting' to describe explicitly some compact moduli spaces of K-polystable log Fano pairs.…

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

Equivalence proven between algebraic stability and geometric stability.

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

We prove the existence and uniqueness of Kähler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler-Ricci flow, and of Keller, Rubinstein on…

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…

New non-Kähler 3-folds constructed via log conifold transitions.

Solves modified conjecture for Fano manifolds using Ding stability.

Study geometric properties of log Calabi-Yau manifolds, focusing on Fano manifolds with smooth or two proportional components.

Let $(X,ω)$ be a compact normal Kähler space, with Hodge metric $ω$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical Kähler metrics, we introduce a geodesic metric structure on $\mathcal H_ω(X)$, the space of Kähler potentials, whose completion is the fin…

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 shows K-moduli spaces of curves on quadrics and K3 surfaces match with VGIT quotients.

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…

Proves unique degeneration of log Fano fibration germs.

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…

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 …

Study confirms boundedness of certain singularities in log Fano geometry.

Characterizes K-semistability for log Fano cone singularities.

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

Exposes new connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.

We prove some criteria for uniform K-stability of log Fano pairs. In particular, we show that uniform K-stability is equivalent to $β$-invariant having a positive lower bound. Then we study the relation between optimal destabilization conjecture and the conjectural equivalence between uniform K-stability and K-stabilit…

Study calculates volumes of Fano K-moduli spaces in various dimensions.

Introduces new stability concept for Fano fibrations.

Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.

The notion of asymptotically log Fano varieties was given by Cheltsov and Rubinstein. We show that, if an asymptotically log Fano variety $(X, D)$ satisfies that $D$ is irreducible and $-K_X-D$ is big, then $X$ does not admit Kähler-Einstein edge metrics with angle $2πβ$ along $D$ for any sufficiently small positive ra…

The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements abou…

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 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…

Homological mirror symmetry for toric Fano surfaces using Morse homotopy.

We construct proper good moduli spaces parametrizing K-polystable $\mathbb{Q}$-Gorenstein smoothable log Fano pairs $(X, cD)$, where $X$ is a Fano variety and $D$ is a rational multiple of the anti-canonical divisor. We then establish a wall-crossing framework of these K-moduli spaces as $c$ varies. The main applicatio…

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 metrics on them. In particular, we obtain simple examples of Fano threefolds being…

In this paper, we show that along $\mathbb Q$-Fano fibration, when general fibres, base and central fiber (with at worst Kawamata log terminal singularities)are K-poly stable then there exists a relative Kähler-Einstein metric. We introduce the fiberwise Kähler-Einstein foliation and we mention that the main difficulty…

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…

This paper connects soliton-type metrics with weighted CSCK metrics on Fano manifolds.

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

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…

The paper finds symplectic mapping class relations using pencil pairs.

Solves Tian's stabilization problem for toric Fano manifolds.

Study shows K-moduli spaces connect quartic surfaces to K3 surfaces, verifying predictions and classifying degenerations.

Paper answers Jin and Rubinstein's question about Fano manifolds.