arXiv research

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

Study of line bundles on spherical varieties leads to Calabi-Yau metrics.

We prove a criterion for K-stability of a $\mathbb{Q}$-Fano spherical variety with respect to equivariant special test configurations, in terms of its moment polytope and some combinatorial data associated to the open orbit. Combined with the equivariant version of the Yau-Tian-Donaldson conjecture for Fano manifolds p…

Paper computes stability of Q-Fano spherical varieties using test configurations and Futaki invariants.

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

The paper studies K-stability of spherical varieties and their degenerations.

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…

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

The paper classifies equivariant test configurations for spherical varieties.

Sharp inequalities for Kähler manifolds' systolic invariants are established.

This paper sets a lower bound for sample complexity in inverse reinforcement learning.

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…

New toric Fano manifolds found without extremal Kähler metrics.

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

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

Proof of flow convergence on Fano manifolds.

Solves modified conjecture for Fano manifolds using Ding stability.

Quantizes Kähler-Ricci flow for Fano manifolds.

Compactifies Calabi-Yau to weak Fano manifolds.

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…

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.

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…

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 …

Survey on Kähler-Einstein and weighted solitons on Fano manifolds.

Paper confirms Hamilton-Tian conjecture for specific Sasakian manifolds.

Eta invariant computed for circle bundles over Fano manifolds.

The paper studies how certain solitons on Fano manifolds extend to nearby deformations.

We prove that Kahler-Einstein Fano manifolds with finite automorphism groups form Hausdorff moduli algebraic space with only quotient singularities. We also discuss the limits as Q-Fano varieties which should be put on the boundary of its canonical compactification.

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…

We give a characterization of relative Ding stable toric Fano manifolds in terms of the behavior of the modified Ding functional. We call the corresponding behavior of the modified Ding functional the pseudo-boundedness from below. We also discuss the pseudo-boundedness of the Ding / Mabuchi functional of general Fano …

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…

New proof for curvature and diameter estimates on Fano manifolds.

We prove that any Fano manifold of coindex three admitting nef tangent bundle is homogeneous.

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 paper defines a new condition for Fano manifolds and shows its implications on their asymptotic behavior.

Study shows volume limit for K-semistable Fano manifolds.

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

Paper studies Kähler-Ricci flow convergence on Fano manifolds.

We show a relation between the birational superrigidity of Fano manifold and its slope stability in the sense of Ross-Thomas.

Paper discusses conditions for deforming coupled Kähler-Einstein metrics.

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

In this note we report on examples of 7- and 8-dimensional toric Fano manifolds that are not symmetric and still admit a Kaehler-Einstein metric. This answers a question first posed by V.V. Batyrev and E. Selivanova. The examples were found in the classification of toric Fano manifolds up to dimension 8 obtained by M. …

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…

The purpose of this paper is to clarify all of the uniformly relatively Ding stable toric Fano threefolds and fourfolds as well as unstable ones. The key player in our classification result is the Mabuchi constants, which can be calculated by combinatorial data of the associated moment polytopes due to the work of Yao …

We construct a geometrically compactified moduli algebraic space of Kahler-Einstein Fano manifolds.

The interpretation, due to T. Mabuchi, of the classical Futaki invariant of Fano toric manifolds is extended to the case of the Generalized Futaki invariant, introduced by W. Ding and G. Tian, of almost Fano toric varieties. As an application it is shown that the real part of the Generalized Futaki invariant is positiv…

In this short note, based on the work of Wang-Zhu, we determine the greatest lower bounds on Ricci curvature for all toric Fano manifolds.