arXiv research

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

The paper calculates delta invariants for specific geometric structures.

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…

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

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

Defines a new metric on Fano Kaehler-Ricci solitons.

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

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

Classifies K-stable Fano varieties and finds new examples.

Reductive quotients preserve klt singularities in algebraic geometry.

As a generalization of Kahler-Einstein metrics for Fano manifolds with nonvanishing Futaki invariant, Mabuchi solitons are critical points of a Calabi-type energy functional. We study their existence on toric Fano varieties and the underlying algebraic stability notion: relative Ding stability. As a toy model for a YTD…

New Kähler metrics generalize Calabi's and relate to Fano manifolds.

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

The paper examines conditions for Kähler-Einstein metrics on deformations of Fano manifolds.

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…

Study of Kähler-Ricci flow on toric Fano varieties with preserved symplectic condition.

We provide a significant extension of the twisted connected sum construction of G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Some of the main contributions of t…

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

We show that the complex projective space has maximal degree (volume) among all n-dimensional Kahler-Einstein Fano manifolds admitting a holomorphic C^*-action with a finite number of fixed points. The toric version of this result, translated to the realm of convex geometry, thus confirms Ehrhart's volume conjecture fo…

Fano varieties get Mabuchi solitons if they have extremal Kähler metrics.

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 a necessary and sufficient condition in terms of the barycenters of a collection of polytopes for existence of coupled Kähler-Einstein metrics on toric Fano manifolds. This confirms the toric case of a coupled version of the Yau-Tian-Donaldson conjecture. We also obtain a necessary and sufficient condition for…

We introduce an inductive argument for proving birational superrigidity and K-stability of singular Fano complete intersections of index one, using the same types of information from lower dimensions. In particular, we prove that a hypersurface in $\mathbb{P}^{n+1}$ of degree $n+1$ with only ordinary singularities of m…

New examples found of complex manifolds with special metrics.

We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, generalizing Kähler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler-Einstein Fano. In the Fano case we also prove that existe…

For $φ$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $\int_X e^{-φ}.$ We prove that the logarithm of the volume is concave along bounded geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes from…

This is a continuation of paper \cite{Li}. On any toric Fano manifold, we discuss the behavior of limit metric of a sequence of metrics, which are solutions to a continuity family of complex Monge-Ampere equations in Kahler-Einstein problem. We show that the limit metric satisfies a singular complex Monge-Ampere equati…

New insights into Kähler Ricci solitons and Calabi-Yau cones.

New stability thresholds detect K-stability in Fano manifolds.

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

Paper proves various types of varieties minimize a specific energy.

We partially confirm a conjecture of Donaldson relating the greatest Ricci lower bound $R(X)$ to the existence of conical Kahler-Einstein metrics on a Fano manifold $X$. In particular, if $D\in |-K_X|$ is a smooth simple divisor and the Mabuchi $K$-energy is bounded below, then there exists a unique conical Kahler-Eins…

New stability criterion for Fano manifolds using anticanonically balanced metrics.

The paper shows how certain complex projective varieties can be broken down into simpler types.

We consider canonical metrics on Fano manifolds. First we introduce a norm-type functional on Fano manifolds, which has Kahler-Einstein or Kahler-Ricci soliton as its critical point and the Kahler-Ricci flow can be viewed as its (reduced) gradient flow. We then obtain a natural lower bound of this functional. As an app…

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. …

Study of tangent bundle positivity on complex projective varieties.

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…

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…

The paper extends metrics and solitons on toric Fano manifolds with irregular Sasaki-Einstein metrics.

New methods for computing volumes and constructing Fano fibrations.

Proves properness of K-moduli spaces for Fano varieties.

We introduce a flow of Kähler structures over Fano manifolds with formal limit at infinite time a Kähler-Ricci soliton. This flow correspond to a Perelman's modified backward Kähler-Ricci type flow that we call Soliton-Kähler-Ricci flow. It can be generated by the Soliton-Ricci flow. We assume that the Soliton-Ricci fl…

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

New toric Fano manifolds found without extremal Kähler metrics.

The study classifies complex smooth Fano varieties with large pseudoindex.

Constructs symplectic structures from rational functions on fans.

New approach proves K-stability of Fano varieties.