[This question is looking at the paper

• Yau, S.-T., On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I, Comm. Pure Appl. Math., 31 (1978) 339-411, doi:10.1002/cpa.3160310304, (pdf)]

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).

[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", Comm. Pure Appl. Math., 31 (1978) 339-411, doi:10.1002/cpa.3160310304, MR0480350, Zbl 0369.53059.]

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).