On pluricanonical locally conformally almost K\"ahler metrics

TL;DR

This paper studies pluricanonical locally conformally almost K"ahler metrics on almost complex manifolds, generalizing known results for K"ahler and LCK metrics, and explores their geometric properties and implications.

Contribution

It extends properties of LCK metrics to pluricanonical LCAK metrics, providing new characterizations and analyzing their geometric constraints on compact manifolds.

Findings

Pluricanonical LCAK metrics have a fundamental 2-form as an eigenform of the Hodge Laplacian.

On compact pluricanonical LCAK manifolds with non-trivial Lee form, compatible symplectic forms do not exist.

The pluricanonical LCAK condition implies the Lee form is parallel in certain cases.

Abstract

On an almost complex manifold $(M,J)$, a pluricanonical locally conformally almost K\"ahler (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 $\{\theta^\sharp,J\theta^\sharp\}$, where $\theta$ is the Lee form and $D$ is the Levi-Civita connection. On a compact complex manifold, pluricanonical locally conformally K\"ahler (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 $\theta^\sharp$ being real holomorphic, proving in that case $D\theta=0$ when the metric is Gauduchon.