arXiv research

Proves unique degeneration of log Fano fibration germs.

Introduces new stability concept for Fano fibrations.

New methods for computing volumes and constructing Fano fibrations.

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…

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

Inspired by the work of Gross on topological Mirror Symmetry we construct candidate Lagrangian torus fibration models for the 105 families of smooth Fano threefolds. We prove, in the case the second Betti number is one, that the total space of each fibration is homeomorphic to the expected Fano threefold, and show that…

Study Fano fibrations and Kähler-Einstein metrics on their bases.

Study Fano fibrations and Kähler-Ricci flow singularities, proving diameter bounds and curvature estimates.

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

Simply-connected shrinking Kähler-Ricci solitons are proven.

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

The study of projective varieties with nef anticanonical divisors and log terminal singularities.

We study finite-time collapsing limits of the continuity method. When the continuity method starting from a rational initial Kähler metric on a projective manifold encounters a finite-time volume collapsing, this projective manifold admits a Fano fibration over a lower dimensional base. In this case, we prove the conti…

Study confirms boundedness of certain singularities in log Fano geometry.

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.

Homological mirror symmetry for toric Fano surfaces using Morse homotopy.

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

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…

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…

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…

Proves K-polystability for Kähler-Ricci shrinkers with decaying curvature.

We show that if on a compact Kahler threefold there is a solution of the Kahler-Ricci flow which encounters a finite time collapsing singularity, then the manifold admits a Fano fibration. Furthermore, if there is finite time extinction then the manifold is Fano and the initial class is a positive multiple of the first…

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.…

Solves Tian's stabilization problem for toric Fano manifolds.

Paper proves various types of varieties minimize a specific energy.

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.

Log-symplectic structures are Poisson structures $π$ on $X^{2n}$ for which $\bigwedge^n π$ vanishes transversally. By viewing them as symplectic forms in a Lie algebroid, the $b$-tangent bundle, we use symplectic techniques to obtain existence results for log-symplectic structures on total spaces of fibration-like maps…

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

Uniform K-stability of Calabi-Yau fibrations linked to base curve stability.

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 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…