On pluricanonical locally conformally almost Kähler metrics

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 $(F,\theta )$ of the first kind, characterized by the fact that $D\theta$ is $J$-anti-invariant and that the image of the Nijenhuis tensor is $g$-orthogonal to the distribution spanned by $\{{\theta }^{♯},J{\theta }^{♯}\}$, where $\theta$ 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 ${\theta }^{♯}$ being real holomorphic, proving in that case $D\theta =0$ when the metric is Gauduchon.

Minor improvements. Extends the 4D pluricanonical characterization to all dimensions. Proves that for n>4, the pluricanonical condition is equivalent to the fundamental 2-form being a Laplacian eigenform