Generalized locally conformally kaehlerian

Question

Let $M$ be a complex manifold, then a non-degenerated form $\omega \in \Lambda^{1,1}(TM)$ is generalized locally conformally kaehlerian if:

$$d_{\theta}(\omega)=d\omega +\theta \wedge \omega =\theta \wedge d\alpha=$$

$$=(1/2)d_{\theta}\circ d_{-\theta}(\alpha)$$

with $d\theta=0$, and $\theta,\alpha \in \Lambda^1(TM)$.

Have we a geometric interpretation of such a definition?

If $\omega$ and $\omega'$ are LCK, when $\omega+\omega'$ is LCK?

Comments

How well versed are you in Symplectic Topology/Geometry?