arXiv research

New equations derived for Kähler metrics, linking stability and curvature.

The paper introduces new equations in Kähler geometry and proves their solutions and convexity.

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

Study Kähler metrics with constant scalar curvature using coupled equations.

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

Uniform bounds derived for fully non-linear equations.

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…

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

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 on weighted cscK metrics on Kähler varieties with singularities.

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…

Study of Hitchin moduli spaces over Teichmüller space.

In this paper, we derive estimates for scalar curvature type equations with more singular right hand side. As an application, we prove Donaldson's conjecture on the equivalence between geodesic stability and existence of cscK when $Aut_0(M,J)\neq0$. Moreover, we also show that when $Aut_0(M,J)\neq0$, the properness of …

The scalar curvature equation for rotation invariant Kähler metrics on $\mathbb{C}^n \backslash \{0\}$ is reduced to a system of ODEs of order 2. By solving the ODEs, we obtain complete lists of rotation invariant zero or positive csck on $\mathbb{C}^n \backslash \{0\}$ in lower dimensions. We also prove that there doe…

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…

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.

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

We first define Pseudo-Calabi flow, as {equation*} {{aligned}{{\partial \varphi}\over {\partial t}}&= -f(\varphi), \triangle_varphi f(\varphi) &= S(\varphi) - \ul S.{aligned}. \end{equation*} Then we prove the well-posedness of this flow including the short time existence, the regularity of the solution and the continu…

The aim of this paper is to investigate uniqueness of conic constant scalar curvature Kaehler (cscK) metrics, when the cone angle is less than $π$. We introduce a new Hölder space called $\cC^{4,\a,\b}$ to study the regularities of this fourth order elliptic equation, and prove that any $\cC^{2,\a,\b}$ conic cscK metri…

Proves existence of weighted-cscK metrics on Kähler manifolds.

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

Study shows convergence of cscK surfaces in Hilbert scheme.

In this paper we study constant scalar curvature equation (CSCK), a nonlinear fourth order elliptic equation, and its weak solutions on Kähler manifolds. We first define a notion of weak solution of CSCK for an $L^\infty$ Kähler metric. The main result is to show that such a weak solution (with uniform $L^\infty$ bound…

The article describes canonical metrics on holomorphic fibre bundles.

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

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…

Paper finds explicit optimal lower bound for J-functional in Kähler geometry.

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.

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

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…

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.

Study weighted constant scalar curvature on non-compact toric fibrations, proving K-stability conditions.

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

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

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…

Study on Kähler metrics with curvature constraints.

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.

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…

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.

Solves Dirac equation coupled to vector bundles.

The paper studies finite TYCZ expansions on Kaehler manifolds and their relation to cscK metrics.