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

TL;DR

This paper provides a unified approach to Hodge theory variations using the Lefschetz star operator, connecting classical results with Timorin's T-Hodge theory on differential forms divided by wedge products of Kähler forms.

Contribution

It introduces a variation formula for the Lefschetz star operator that unifies existing Hodge theory results with T-Hodge theory, enhancing understanding of differential forms divided by wedge products.

Findings

Unified approach to classical Hodge theory results

Connection between Lefschetz star operator and T-Hodge theory

Variation formula for the Lefschetz star operator

Abstract

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.