关于 pluricanonical 局部 conformally 近 Kähler 度量
On pluricanonical locally conformally almost Kähler metrics
AI总结：
研究 pluricanonical 局部 conformally 近 Kähler 度量的性质，证明其 Lee 形与度量的关系，并探讨其与辛结构的关联。
摘要：
在几乎复结构 (M,J) 上，pluricanonical 局部 conformally 近 Kähler (LCAK) 度量 g 由第一类局部 conformally 拓扑结构 (F,θ) 诱导，其特征是 Dθ 是 J-反不变的，并且 Nijenhuis 张量的像与由 {θ^♯,Jθ^♯} 张成的分布正交。在紧复流形上，pluricanonical 局部 conformally Kähler (LCK) 度量具有平行的 Lee 形。同样结论适用于 LCK Chern-Ricci 平坦 Gauduchon 度量。我们推广这两个结果到 LCAK 度量。此外，我们观察到在紧 pluricanonical LCAK 流形上，若 Lee 形非平凡，则不存在与相同几乎复结构相容的辛形式。并且我们指出 pluricanonical LCAK 条件意味着基本 2-形式是调和拉普拉斯算子的特征形式，并在紧流形上给出其简单刻画。最后，我们研究 Lee 形为实 holomorphic 的 LCAK 度量，证明在 Gauduchon 度量下 Dθ=0。
英文摘要：
On an almost complex manifold $(M,J)$, a pluricanonical locally conformally almost Kähler (LCAK) metric $g$ is induced by a locally conformally symplectic structure $(F,θ)$ of the first kind, characterized by the fact that $Dθ$ is $J$-anti-invariant and that the image of the Nijenhuis tensor is $g$-orthogonal to the distribution spanned by $\{θ^\sharp,Jθ^\sharp\}$, where $θ$ is the Lee form and $D$ is the Levi-Civita connection. On a compact complex manifold, pluricanonical locally conformally Kähler (LCK) metrics have parallel Lee form. The same conclusion holds for LCK Chern--Ricci flat Gauduchon metrics. We generalize both results to LCAK metrics. We also observe that on a compact pluricanonical LCAK manifold with a non-trivial Lee form, there is no symplectic form compatible with the same almost complex structure. Moreover, we remark that the pluricanonical LCAK condition implies that the fundamental $2$-form is an eigenform of the Hodge Laplacian, and we give a simple characterization of the pluricanonical LCAK condition on compact manifolds. Finally, we study LCAK metrics with $θ^\sharp$ being real holomorphic, proving in that case $Dθ=0$ when the metric is Gauduchon.
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