The Connection 1-Form on Hermitian Manifolds

This article studies Hermitian structure on complex manifolds. We introduce the torsion of the Chern connection on Hermitian manifolds and present a global result with the K？hler form related to the torsion coefficients with respect to the connection 1-form. On compact Hermitian manifolds, by introducing a linear operator and, based on it, establishing a duality of differential operators on $\left(p,q\right)$ -forms, we prove an identity between the K？hler form and the connection 1-forms. Furthermore, we show that the integral of the K？hler form can be explicitly expressed via the connection 1-form and vanishes under the Gauduchon condition.