Notes on variation of Lefschetz star operator and T-Hodge theory

These notes were written to serve as an easy reference for {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 {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 u), where T is a finite wedge product of Kähler forms.