  These notes were written to serve as an easy reference for \cite{Wang-AF}.
All the results in this presentation are well-known (or quasi-well-known)
theorems in Hodge theory. Our main purpose was to give a unified approach based
on a variation formula of the Lefschetz star operator, following \cite{Wang-k}.
It fits quite well with Timorin's $T$-Hodge theory, i.e. the Hodge theory on
the space of differential forms divided by $T$ (i.e. forms like $T\wedge u$),
where $T$ is a finite wedge product of K\"ahler forms.
