arXiv subjects

King Leung Lee

Publications and source records attributed to King Leung Lee.

4 recordsLinked to original sources

arXiv2026

On the blow-up formula of the Chow weights for polarized toric manifolds

Let $X$ be a smooth projective toric variety, and let $\widetilde{X}$ denote the blow-up of $X$ at finitely many distinct torus-invariant points. In this paper, we derive an explicit combinatorial formula for the Chow weight of $\widetilde{X}$ in terms of the underlying toric manifold $X$ and the symplectic cuts of its associated Delzant polytope. As an application, we study toric blow-ups of the projective plane and compare their Chow stability with that of blow-ups at general points.

math.AG↗

arXiv2023

Asymptotic Chow stability of symmetric reflexive toric varieties

In this note, we study the asymptotic Chow stability of toric varieties. We provide examples of symmetric reflexive toric varieties that are not asymptotic Chow semistable. On the other hand, we also show that any weakly symmetric reflexive toric varieties which have regular triangulation (special) are asymptotic Chow polystable. After that, we provide another criteria that can show a symmetric reflexive toric variety is asymptotic Chow polystable. In particular, we give two examples that are asymptotic Chow polystable, but not special. We also provide some examples of special polytopes, mainly in 2 or 3 dimensions, and some in higher dimensions.

math.AG↗

arXiv2023

Moment map for coupled equations of Kähler forms and curvature

In this paper we introduce two new systems of equations in Kähler geometry: The coupled p equation and the generalized coupled cscK equation. We motivate the equations from the moment map pictures, prove the uniqueness of solutions and find out the obstructions to the solutions for the second equation. We also point out the connections between the coupled cscK equation, the coupled Kähler Yang-Mills equations and the deformed Hermitian Yang-Mills equation. Moreover, using this moment map, we can show the Mabuchi functional for the generalized coupled cscK equation, and a special case of the coupled Kähler Yang-Mills equations and the deformed Hermitian Yang-Mills equation are convex along the smooth geodesic, which is different from the one using the moment map picture from the gauge group. In our case, the geodesic is given by the natural metric on the product of smooth Kähler potential $\mathcal{K}(X,ω_0)\times \cdots \times \mathcal{K}(X,ω_k)$.

math.DG↗

arXiv2022

Moment map, convex function and extremal point

The moment map $μ$ is a central concept in the study of Hamiltonian actions of compact Lie groups $K$ on symplectic manifolds. In this short note, we propose a theory of moment maps coupled with an $\mathrm{Ad}_K$-invariant convex function $f$ on $\mathfrak{k}^{\ast}$, the dual of Lie algebra of $K$, and study the properties of the critical point of $f\circμ$. Our motivation comes from Donaldson \cite{Donaldson2017} which is an example of infinite dimensional version of our setting. As an application, we interpret Kähler-Ricci solitons as a special case of the generalized extremal metric.

math.DG↗