AI 中文总结
该研究证明射影化空间 $\boldsymbol{P}(E)$ 存在对合型兼容极端近凯勒度量等价于存在Calabi极端凯勒度量，利用对应关系简化方程，并应用于秩4向量丛的射影丛，证明其为Calabi梦想流形。
AI 中文摘要
设紧复曲线 $C$ 上的全纯向量丛 $E$ 可分解为稳定向量丛的直和。对于射影化空间 $\boldsymbol{P}(E)$，我们证明：存在Lejmi意义下的对合型兼容极端近凯勒（aK）度量，等价于存在Calabi极端凯勒度量。该结果基于作者与Yin在前期工作中证明的、关于 $\boldsymbol{P}(E)$ 的矩多面体 $\boldsymbol{\triangle}$ 的Yau-Tian-Donaldson对应关系。其主要优势在于，对合型兼容极端近凯勒度量是二阶线性偏微分方程的解，而非Calabi极端凯勒度量对应的四阶非线性偏微分方程的解。作为应用，我们证明：当 $E$ 的秩为4，且 $C$ 为椭圆曲线或射影直线时，$\boldsymbol{P}(E)$ 是Calabi梦想流形，即其在每个凯勒类中都存在极端凯勒度量。
英文摘要
Consider a holomorphic vector bundle $E$ over a compact complex curve $C$ which decomposes as a sum of stable vector bundles. For the projectivization $\mathbb{P}(E)$, we prove that the existence of a compatible extremal almost Kähler (aK) metric of involutive type in the sense of Lejmi is equivalent to the existence of a Calabi extremal Kähler metric. This result rests on the Yau--Tian--Donaldson correspondence in terms of the moment polytope $Δ$ for $\mathbb{P}(E)$, proved by the author and Yin in a previous work. The main advantage is that compatible extremal aK metrics of involutive type are solutions to a second-order linear PDE, rather than a fourth-order nonlinear PDE for Calabi's extremal Kähler metrics. As an application, we prove that when $E$ has rank $4$ and $C$ is an elliptic curve or the projective line, $\mathbb{P}(E)$ is a Calabi dream manifold, i.e. admits an extremal Kähler metric in every Kähler class.
