arXiv research

The paper calculates delta invariants for specific geometric structures.

Characterizes K-semistability for log Fano cone singularities.

Proves boundedness of log Fano cone singularities with bounded local volumes.

Smoothness proven for conical Calabi-Yau potentials on Fano cones.

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…

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

Study shows Futaki invariant vanishes on most Fano threefolds.

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

Study confirms boundedness of certain singularities in log Fano geometry.

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

This is the third and final paper in a series which establish results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2π. We also put all our technical results together to complete the proof of the…

The continuity method is used to deform the cone angle of a weak conical Kähler-Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldson's Openness Theorem on deforming cone angle \cite{Don} by combining it with the regulari…

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…

Exposes new connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.

Given a klt singularity $x\in (X, D)$, we show that a quasi-monomial valuation $v$ with a finitely generated associated graded ring is the minimizer of the normalized volume function $\widehat{\rm vol}_{(X,D),x}$, if and only if $v$ induces a degeneration to a K-semistable log Fano cone singularity. Moreover, such a mi…

In this paper, we prove Matsushima's theorem for Kähler-Einstein metrics on a Fano manifold with cone singularities along a smooth divisor that is not necessarily proportional to the anti-canonical class. We then give an alternative proof of uniqueness of Kähler-Einstein cone metrics by the continuity method. Moreover,…

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…

The paper studies Kähler compactifications of C^n and their applications in geometry.

New invariants detect Fano varieties' K-stability.

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

We prove that on one Kähler-Einstein Fano manifold without holomorphic vector fields, there exists a unique conical Kähler-Einstein metric along a simple normal crossing divisor with admissible prescribed cone angles. We also establish a curvature estimate for conic metrics along a simple normal crossing divisor which …

This is the first of a series of three papers which provide proofs of results announced recently in arXiv:1210.7494.

We present explicit constructions of complete Ricci-flat Kahler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Ka…

We prove that if a $\mathbb{Q}$-Fano variety $V$ specially degenerates to a Kähler-Einstein $\mathbb{Q}$-Fano variety $V$, then for any ample Cartier divisor $H=-r^{-1} K_V$ with $r\in \mathbb{Q}_{>0}$, the normalized volume $\widehat{\rm vol}(v)=A_{\mathcal{C}}^n(v)\cdot {\rm vol}(v)$ is globally minimized at the cano…

Study limits of Kähler-Einstein metrics with cone singularities on complex projective manifolds.

This is the second of a series of three papers which provide proofs of results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle is less than 2π. We show that these are in a natrual way projective algebraic var…

In this paper, we study the stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle $2πβ$ along the divisor, then for any $β'$ sufficiently close to $β$, the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein me…

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

The paper constructs ALF Calabi-Yau metrics on specific manifolds.

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

It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof exploits convexity properties of the Ding functiona…

We give a criterion for the coercivity of the Mabuchi functional for general Kähler classes on Fano manifolds in terms of Tian's alpha invariant. This generalises a result of Tian in the anti-canonical case implying the existence of a Kähler-Einstein metric. We also prove the alpha invariant is a continuous function on…

The study examines conical Kähler-Einstein metrics on specific Fano manifolds and their behavior at the limit angle.

In this article, we systematically investigate the stability properties of certain warped product Einstein manifolds. We characterize stability of these metrics in terms of an eigenvalue condition of the Einstein operator on the base manifold. In particular, we prove that all complete manifolds carrying imaginary Killi…

Maps Kähler cones to moduli spaces of stable manifolds.

We show that the existence of constant scalar curvature Kähler (cscK) metrics with cone singularities is equivalent to the properness of log $K$-energy. We also prove their equivalence to the geodesic stability. They are extensions of the solution of the properness conjecture and Donaldson's geodesic stability conjectu…

We give some non-existence results for Kähler-Einstein metrics with conical singularities along a divisor on Fano manifolds. In particular we show that the maximal possible cone angle is in general smaller than the invariant R(M). We study this discrepancy from the point of view of log K-stability.

Researchers prove existence of Kähler-Einstein metrics with conic singularities on Fano manifolds.

We construct gradient Kähler-Ricci solitons on Ricci-flat Kähler cone manifolds and on line bundles over toric Fano manifolds. Certain shrinking and expanding solitons are pasted together to form eternal solutions of the Ricci flow. The method we employ is the Calabi ansatz over Sasaki-Einstein manifolds, and the resul…

We study the singularities of Legendrian subvarieties of contact manifolds in the complex-analytic category and prove two rigidity results. The first one is that Legendrian singularities with reduced tangent cones are contactomorphically biholomorphic to their tangent cones. This result is partly motivated by a problem…

Paper proves existence of unique constant scalar curvature Kähler metric under certain conditions.

We study the scalar curvature of Kähler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of Kähler--Einstein Fano …

This is a continuation to the paper [arXiv:1511.08164] in which a problem of minimizing normalized volumes over $\mathbb{Q}$-Gorenstein klt singularities was proposed. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler-Einstein metrics on Fano varieties. In particul…

In this paper, we consider the twisted Kähler-Ricci soliton, and show that the existence of twisted Kähler-Ricci soliton with semi-positive twisting form is closely related to the properness of some energy functionals. We also consider the conical Kähler-Ricci soliton, and obtain some existence results. In particular, …

New invariants prove existence of Kahler-Einstein metrics on big classes.

Extremal Kahler metrics and Sasaki-Einstein metrics characterized via coercive energy.

Over a compact Kähler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a Kähler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature Kähler metrics with conic singularities: existence result under small deformatio…

We describe two simple obstructions to the existence of Ricci-flat Kahler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kahler-Einstein metrics on Fano orbifolds.…