On pluricanonical locally conformally almost Kähler metrics
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910076585902080 |
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| author | Addison, Ethan Draghici, Tedi Lejmi, Mehdi |
| author_facet | Addison, Ethan Draghici, Tedi Lejmi, Mehdi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12352 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On pluricanonical locally conformally almost Kähler metrics Addison, Ethan Draghici, Tedi Lejmi, Mehdi Differential Geometry 53C55 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. |
| title | On pluricanonical locally conformally almost Kähler metrics |
| topic | Differential Geometry 53C55 |
| url | https://arxiv.org/abs/2602.12352 |