arXiv research

Paper confirms Hamilton-Tian conjecture for specific Sasakian manifolds.

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

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

New methods for computing volumes and constructing Fano fibrations.

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

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…

In this note, using the recent compactness results of Tian and Chen-Donaldson-Sun, we prove the K-semistable version of Yau-Tian-Donaldson correspondence for Fano manifolds.

We survey recent results on the existence of Kähler-Einstein metrics on certain smoothable Fano varieties, focusing on the importance of such metrics in the construction of compact algebraic moduli spaces of K-polystable Fano varieties. Moreover, we give some applications and we discuss some natural problems which dese…

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

Study of generalized almost-Kähler-Ricci solitons and their implications.

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…

We obtain a compactness result for Fano manifolds and Kähler Ricci flows. Comparing to the more general Riemannian versions by Anderson and Hamilton, in this Fano case, the curvature assumption is much weaker and is preserved by the Kähler Ricci flows. One assumption is the boundedness of the Ricci potential and the ot…

We provide a direct proof of time-slice weak compactness along the Kähler Ricci flow on Fano manifolds.

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

Study Kähler-Ricci flow on manifolds with singularities.

Paper studies Kähler-Ricci flow convergence on Fano manifolds.

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…

Kahler-Einstein metrics linked to eigenvalue gaps on Fano manifolds.

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

The paper proves uniqueness and existence of Kähler-Einstein metrics on certain compactifications.

New stability thresholds detect K-stability in Fano manifolds.

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

We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G_2 developed in math.DG/0012189. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors, the latter `matching' via a certain non-holomorphic map. Suitable examples of threefolds …

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.…

We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $\barpartial$-Laplacian on compact Kähler manifolds. This situation typically occurs for a sequence of Fano manifolds with anticanonical Kähler class. We apply it to show that, if an almost smooth Fano-Ricci limit space admits a Kähler…

We show that if a Fano manifold $M$ is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then $M$ admits a Kähler-Einstein metric. This is a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-Donaldson-Sun, and can be used to ob…

Constructs moduli spaces for Calabi-Yau cones and Sasaki-Einstein manifolds.

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. …

In this short note we announce a regularity theorem for Kähler-Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of Kähler-Ricci flow on Fano 3-manifolds. Moreover, we also present a partial $C^0$ estimate to the Kähler-Ricci flow under …

We classify compact Kähler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex $3$-folds of the form $P(T^*X)$ whose tangent bundles are nef. Moreover, we show that if $X$ is a Fano manifold such t…

We prove the existence of Kahler-Einstein metric on a K-stable Fano manifold using the recent compactness result on Kahler-Ricci flows. The key ingredient is an algebro-geometric description of the asymptotic behavior of Kahler-Ricci flow on Fano manifolds. This is in turn based on a general finite dimensional discussi…

In this paper, we study the behavior of Ricci flows on compact orbifolds with finite singularities. We show that Perelman's pseudolocality theorem also holds on orbifold Ricci flow. Using this property, we obtain a weak compactness theorem of Ricci flows on orbifolds under some natural technical conditions. This genera…

We give a differential-geometric construction of Calabi-Yau fourfolds by the `doubling' method, which was introduced in \cite{DY14} to construct Calabi-Yau threefolds. We also give examples of Calabi-Yau fourfolds from toric Fano fourfolds. Ingredients in our construction are \emph{admissible pairs}, which were first d…

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…

We prove that on Fano manifolds, the Kähler-Ricci flow produces a "most destabilising" degeneration, with respect to a new stability notion related to the H-functional. This answers questions of Chen-Sun-Wang and He. We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the su…

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

Estimates quantum cohomology complexity for Fano varieties and homogeneous spaces.

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…

In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tange…

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

Proves invariance of weighted extremal Kähler metrics under smooth blowups.

Proves properness of K-moduli spaces for Fano varieties.

We give a new, connected-sum-like construction of Riemannian metrics with special holonomy G_2 on compact 7-manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian manifolds with holonomy SU…

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.