# Harmonic symmetries for Hermitian manifolds

## TL;DR

This paper explores harmonic differential forms on Hermitian manifolds, revealing their symmetries, dualities, and a representation of sl(2,C) that generalizes known structures from Kähler geometry, with implications for topology.

## Contribution

It introduces a new framework for understanding harmonic forms on Hermitian manifolds, extending classical Kähler results through sl(2,C) representations.

## Key findings

- Harmonic forms satisfy Serre, Hodge, and conjugation dualities.
- Hard Lefschetz duality is established via sl(2,C) representation.
- Topological implications for Hermitian manifolds are derived.

## Abstract

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a representation of $sl(2,\mathbb{C})$, generalizing the well known structure on the harmonic forms of compact K\"ahler manifolds. Some topological implications are deduced.