# A problem of the volume form of Kähler manifold in the paper of Yau's proof of Calabi conjecture

[This question arises from a look at the paper

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).

• I think you expand out and integrate by parts to show that the integral is the same as for $\Omega$. In fact, the homology class of $\Omega+d\bar\partial \varphi$ is clearly that of $\Omega$. Mar 29 at 18:28
• Thanks for answering, I just started learning Kahler geometry while reading this paper. You mean that $\Omega+d \bar{\partial} \varphi$ = $\Omega+\partial \bar{\partial} \varphi$ then integraing $\Omega+\partial \bar{\partial} \varphi$ equals to integrating $\Omega$(Since $M$ has no boundary and Stokes), then similarly expand out $(\Omega+\partial \bar{\partial} \varphi)^{m}$ and integrate by parts? Mar 30 at 10:15
• yes, that's right. Mar 30 at 10:22

## 1 Answer

Just to close this off: note that $$d=\partial+\bar\partial$$ and that $$\partial^2=0$$ so $$\partial\bar\partial=d\bar\partial$$, and therefore $$\Omega+\partial\bar\partial\varphi=\Omega+d\bar\partial\varphi$$ is in the same cohomology class as $$\Omega$$. Since wedge product of forms descends to the usual product in cohomology, $$(\Omega+d\bar\partial\varphi)^n=\Omega^n$$ in cohomology, giving the same volume integral over our compact manifold. On the other hand, since the Monge-Ampere equation is elliptic, scalar, determined, it is locally solvable, so $$\Omega+d\bar\partial\varphi$$ can achieve any multiple of any given volume form, locally, by suitable local choice of $$\varphi$$. So we cannot guaranteed that $$\Omega+d\bar\partial\varphi$$ is not zero somewhere, if we allow arbitrary choice of $$\varphi$$. So we can't be sure that this $$(\Omega+\partial\bar\partial\varphi)^n$$ is actually a volume form, i.e. a nowhere-zero top-degree form with positive integral. That requires more information.