arXiv research

Study shows convergence of cscK surfaces in Hilbert scheme.

Study moduli space of cscK surfaces around toric ones, introducing foldable surfaces.

Study on Kähler metrics on ruled surfaces, proving existence and non-existence.

Let X be a compact toric surface. There exists a sequence of torus equivariant blow-ups of X such that the blown-up toric surface obtained admits a cscK metric.

The paper finds conical higher cscK metrics on minimal ruled surfaces with conical singularities.

The aim of this article is to study expansions of solutions to an extremal metric type equation on the blow-up of constant scalar curvature Kähler surfaces. This is related to finding constant scalar curvature Kähler (cscK) metrics on K-stable blow-ups of extremal Kähler surfaces

Study asymptotic behavior of second Chern forms on degenerating Kähler-Einstein surfaces.

New system modifies constant scalar curvature Kähler condition with a 'Higgs field'.

Constructs a new type of metric for elliptic surfaces.

In this note we discuss the problem of resolving conically singular cscK varieties to construct smooth cscK manifolds, showing a glueing result for (some) crepant resolutions of cscK varieties with discrete automorphism groups.

We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope $μ$ for a projective manifold and for each of its subschemes, and show that if $X$ is cscK then $μ(Z)\leμ(X)$ for all subschemes $Z$. This gives man…

Proves existence of weighted-cscK metrics on Kähler manifolds.

In this article, I prove the following statement: Every compact complex surface with even first Betti number is deformation equivalent to one which admits an extremal Kähler metric. In fact, this extremal Kähler metric can even be taken to have constant scalar curvature in all but two cases: the deformation equivalence…

The paper proves the existence of singular cscK metrics on smoothable varieties.

We discuss a natural extension of the Kähler reduction of Fujiki and Donaldson, which realises the scalar curvature of Kähler metrics as a moment map, to a hyperkähler reduction. Our approach is based on an explicit construction of hyperkähler metrics due to Biquard and Gauduchon. This extension is reminiscent of how o…

Study Kähler metrics with constant scalar curvature using coupled equations.

The study finds conditions for Kaehler-Einstein and cscK metrics on certain manifold coverings.

In this paper, we consider a CscK metric defined away from divisor and with metric upper bound and lower bound going to zero in certain rate. And we'll prove that this "nicely" behaved metric is a smooth CscK metric across the divisor.

The paper proves conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.

The paper introduces a new system of equations for Hessian-cscK metrics.

Study on weighted cscK metrics on Kähler varieties with singularities.

New invariants help solve existence of weighted cscK metrics.

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 introduce a cohomological obstruction to solving the constant scalar curvature Kähler (cscK) equation twisted by a semipositive form, appearing in works of Fine and Song-Tian. Geometrically this gives an obstruction for a manifold to be the base of a holomorphic submersion carrying a cscK metric in certain ``adiabat…

Researchers prove existence of cscK metrics on smooth minimal models.

Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.

Let $(M, g)$ be a Kaehler manifold whose associated Kaehler form $ω$ is integral and let $(L, h)\rightarrow (M, ω)$ be a quantization hermitian line bundle. In this paper we study those Kaehler manifolds $(M, g)$ admitting a finite TYCZ expansion. We show that if the TYCZ expansion is finite then $T_{mg}$ is indeed a p…

This paper characterizes mu-cscK metrics using Perelman's W-entropy.

Let $(X,ω)$ be a compact Kähler manifold and $\mathcal H$ the space of Kähler metrics cohomologous to $ω$. If a cscK metric exists in $\mathcal H$, we show that all finite energy minimizers of the extended K-energy are smooth cscK metrics, partially confirming a conjecture of Y.A. Rubinstein and the second author. As a…

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

Study on Kähler metrics with curvature constraints.

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature Kähler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture…

This is a continuation of the previous articles on Kahler cone metrics. In this article, we introduce weighted function spaces and provide a self-contained treatment on cone angles in the whole interval $(0,1]$. We first construct geodesics in the space of Kahler cone metrics (cone geodesics). We next determine the ver…

Any constant-scalar-curvature Kaehler (cscK) metric on a complex surface may be viewed as a solution of the Einstein-Maxwell equations, and this allows one to produce solutions of these equations on any 4-manifold that arises as a compact complex surface with b_1 even. It is shown, however, that not all solutions of th…

We partially confirm an old conjecture of Donaldson that if there exists a cscK metrics in a given Kähler class, then there is no degenerated geodesic ray which is tamed by a bounded ambient geometry unless it parallels to a holomorphic line consists of cscK metrics only. We also prove that for simple test configuratio…

Non-Archimedean balanced metrics approximate cscK metrics for totally degenerate abelian varieties

Analytic K-semistability connects curvature to metric existence.

We prove that constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a very recent result by R. Berman, T. Darvas and C. Lu regarding properness of the K-energy, it moreover follows that cscK manif…

Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.

The paper proves the existence of a special type of metric on complex manifolds.

In this paper we compute the Futaki invariant of adiabatic Kaehler classes on resolutions of Kaehler orbifolds with isolated singularities. Combined with previous existence results of extremal metrics by Arezzo-Lena-Mazzieri, this gives a number of new existence and non-existence results for cscK metrics.

Compact metrics found near Kähler manifold's canonical class.

The purpose of this paper is to generalize the convexity of Mabuchi's functional to the conic setting. We first established a frame to study conic cscK metrics, and then the conic Mabuchi functional was introduced in such a way that conic cscK metrics are its critical points. Finally we proved the convexity of the coni…

The paper proves a unique cscK metric for uniformly K-stable Kähler manifolds.

We consider the Kähler-Ricci flow on a Fano manifold. We show that if the curvature remains uniformly bounded along the flow, the Mabuchi energy is bounded below, and the manifold is K-polystable, then the manifold admits a Kähler-Einstein metric. The main ingredient is a result that says that a sufficiently small pert…

Establishes convexity and coercivity of K-energy functional for complex tori.

The paper defines and analyzes Kähler metrics near a compact manifold, showing their deviation from Poincaré-type metrics.

Uniform K-stability of Calabi-Yau fibrations linked to base curve stability.