arXiv research

Study confirms boundedness of certain singularities in log Fano geometry.

Solves Tian's stabilization problem for toric Fano manifolds.

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

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

Proves unique degeneration of log Fano fibration germs.

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

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…

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.

Characterizes K-semistability for log Fano cone singularities.

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

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…

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

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

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…

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…

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

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…

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

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…

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…

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

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…

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…

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…

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

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.…

The paper connects Kähler-Ricci shrinkers to Fano fibrations in algebraic geometry.

Solves modified conjecture for Fano manifolds using Ding stability.

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…

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

Equivalence proven between algebraic stability and geometric stability.

Extends probabilistic approach for Kahler-Einstein metrics on Fano manifolds.

We consider the space KR(n,F) of Kahler-Ricci solitons on n-dimensional Fano manifolds with Futaki invariant bounded by F. We prove a partial C^0 estimate for KR(n,F) as a generalization of the recent work of Donaldson-Sun for Fano Kahler-Einstein manifolds. In particular, any sequence in KR(n,F) has a convergent sub…

In this short paper, we improve the result of Phong-Song-Sturm on degeneration of Fano Kähler-Ricci solitons by removing the assumption on the uniform bound of the Futaki invariant. Let $\mathcal{KR}(n)$ be the space of Kähler-Ricci solitons on $n$-dimensional Fano manifolds. We show that after passing to a subsequence…

Eigenvalue estimates for Hodge Laplacian on Fano manifolds lead to geometric insights.

The paper proves the existence of Sasaki-Einstein metrics on specific Sasakian manifolds.

We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an application, we show that they have a Kaehler-Einstein metric if they are general.

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

Study on GL(2) geometries on complex manifolds, focusing on Kähler-Einstein and Fano manifolds.

The study classifies Fano varieties with specific pseudoindex.

In this paper, we prove the conic version of YTD conjecture on log 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…

New stability criteria for Fano varieties using generalized b-divisors.

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…