On pluricanonical locally conformally almost Kähler metrics
Ethan Addison, Tedi Draghici, Mehdi Lejmi
TL;DR
This work extends pluricanonical LCK-type rigidity to pluricanonical locally conformally almost Kähler metrics on non-integrable almost complex manifolds. It defines the pluricanonical condition via $D\theta$ being $J$-anti-invariant and $\mathrm{Im}\,N$ being $g$-orthogonal to $\mathrm{Span}(\theta^{\sharp},J\theta^{\sharp})$, and derives Bochner-type identities, Gauduchon consequences, and dimension-4 and Lie-algebra characterizations. A key geometric consequence is that on compact non-globally conformally first-kind LCS manifolds, no symplectic form can be compatible with the same almost complex structure; and in Gauduchon, Hermitian-Ricci flat settings, $\theta$ is forced to be $D$-parallel or the metric to be pluricanonical. The paper also provides explicit 4D Lie algebra examples and a complete classification of unimodular almost abelian pluricanonical LCAK 4D Lie algebras, along with simple conditions equating anti-pluricanonical and holomorphic Lee fields. Together, these results generalize Vaisman-type rigidity to the non-integrable LCAK world and supply concrete algebraic models for the pluricanonical condition.
Abstract
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 and 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 pluricanonical 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 give a simple characterization of the pluricanonical LCAK condition on Lie algebras. Finally, we study LCAK metrics with $θ^\sharp$ being real holomorphic, proving in that case $Dθ=0$ when the metric is Gauduchon.