[This question arises from a look at the paper

* Shing-Tung Yau, "[On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I](https://jasonpayne.webs.com/Math5339/On%20the%20Ricci%20Curvature%20of%20a%20Compact%20Kahler%20Manifold%20and%20the%20Complex%20Monge-Ampere%20Equation%20I,%20S.T.%20Yau.pdf)", Comm. Pure Appl. Math., **31** (1978) 339-411, doi:[10.1002/cpa.3160310304](https://doi.org/10.1002/cpa.3160310304), [MR0480350](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0480350), [Zbl 0369.53059](https://zbmath.org/?q=an%3A0369.53059).]

My problem arises from (4.1)

It said that itegrating $(\Omega+\partial \bar{\partial} \varphi)^{m}=(\exp \{F\}) \Omega^{m}(4.1)$ then we get $\int \exp \{F\}=\operatorname{Vol}(M)$ where $\Omega$ is the kahler form.

Does this mean $(\Omega+\partial \bar{\partial} \varphi)^{m}$ is also a volume form? I'm confused this step of integrating (4.1).