arXiv research

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…

A new method studies symplectic configurations in rational 4-manifolds using computer-aided techniques.

Special orthogonal representations from octonions have geometric properties linked to binary cubics.

We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano Kahler-Einstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricci-flat metrics have slower-than-Euclidean volume growth and quadratic curvature decay. …

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…

In this paper we investigate codimension one Fano distributions on Fano manifolds with Picard number one. We classify Fano distributions of maximal index on complete intersections in weighted projective spaces, Fano contact manifolds, Grassmannians of lines and their linear sections, and describe their moduli spaces. A…

Kähler-Ricci flow shows type II singularity on Fano threefolds.

New methods for computing volumes and constructing Fano fibrations.

Proves properness of K-moduli spaces for Fano varieties.

The study examines K-polystability on Fano 4-folds with specific Lefschetz defects.

New toric Fano manifolds found without extremal Kähler metrics.

Paper studies limits of Kähler-Ricci flow on Fano G-manifolds.

The study classifies complex smooth Fano varieties with large pseudoindex.

New approach proves K-stability of Fano varieties.

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

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

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…

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

Proves unique degeneration of log Fano fibration germs.

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

Solves modified conjecture for Fano manifolds using Ding stability.

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…

Study positive characteristic Fano 4-folds with nef tangent bundles.

The paper proves Mabuchi solitons and constants on Fano admissible manifolds.

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

We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular att…

New research confirms Kähler-Einstein metrics for all Fano threefolds of degree 22.

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…

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

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

The study classifies Fano varieties with specific pseudoindex.

In this paper, we prove an existence result for Kähler-Einstein metrics on $\mathbb Q$-Fano compactifications of Lie groups. As an application, we classify $\mathbb Q$-Fano compactifications of $SO_4(\mathbb C)$ which admit a Kähler-Einstein metric with the same volume as that of a smooth Fano compactification of $SO_4…

Short note shows unbounded dimensions in Fano K-moduli spaces.

In this paper, we prove that any solution of Kähler-Ricci flow on a Fano compactification $M$ of semisimple complex Lie group, is of type II, if $M$ admits no Kähler-Einstein metrics. As an application, we found two Fano compactifications of $\mathrm{SO}_4(\mathbb{C})$ and one Fano compactification of $\mathrm{Sp}_4(\m…

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

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 consider Fano manifolds admitting an algebraic torus action with general orbit of codimension one. Using a recent result of Datar and Szekelyhidi, we effectively determine the existence of Kahler-Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kahl…

Kähler-Ricci flow on spherical Fano manifolds converges to a soliton.

Quantizes Kähler-Ricci flow for Fano manifolds.

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…

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

We introduce a new effective stability named "divisorial stability" for Fano manifolds which is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume function. As a corollary, we prove that the first coordinate of the barycenter of the …

Compactifies Calabi-Yau to weak Fano manifolds.

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…

Proves SYZ conjecture for certain toric Fano hypersurfaces.

Homological mirror symmetry for toric Fano surfaces using Morse homotopy.

Study lower bounds on modified K-energy on Fano manifolds with Kähler-Ricci solitons.

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