Now we assume an $n$ dimensional Kähler manifolds $(X,\omega,J)$ admits an $T^{k}(k\geq 1)$ action which is holomorphic and Hamiltonian. We first assume the action is free. Let $(z_{1},\cdots,z_{k})$ be the moment map. Then similar discussion to that in Section 2.1 yields locally a family of Kähler forms $\tilde{\omega}$ on the complex quotient, parametrized by $(z_{1},\ldots z_{k})\in\mathbb{R}^{k}$, a family of connection $1$-forms $-\sqrt{-1}\Theta_{j}(j=1,\cdots,k)$ and a positive definite $k\times k$ real symmetric matrix $W=(W_{ij})$ with the inverse matrix

(2.63)

$$W^{ij}=\langle\partial_{t_{i}},\partial_{t_{j}}\rangle,$$

such that the following system of equations hold

(2.64)

$$\begin{cases}\partial_{z_{j}}\tilde{\omega}=d_{D}\Theta_{j}\\ \partial_{z_{j}}\Theta_{i}=-d^{c}_{D}W_{ij}\\ \partial_{z_{l}}W_{ij}=\partial_{z_{j}}W_{il}.\end{cases}$$

As before the first two equations combine to give an equation on $(\tilde{\omega},W^{ij})$

(2.65)

$$\partial_{z_{i}}\partial_{z_{j}}\tilde{\omega}+d_{D}d_{D}^{c}W_{ij}=0.$$

Now suppose the complex quotient $D$ is Calabi-Yau with a holomorphic volume form $\Omega_{D}$, then the Calabi-Yau equation on $X$ becomes

(2.66)

$$\frac{\tilde{\omega}^{n-k}}{(n-k)!}=\frac{(\sqrt{-1})^{(n-k)^{2}}}{2^{n-k}}\det(W_{ij})\cdot\Omega_{D}\wedge\bar{\Omega}_{D}.$$

This equation has been derived by Matessi [Mat01] and Zharkov [Zha04]. Again when the $T^{k}$ action is not free one should replace (2.65) by a distributional equation. We will discuss a simplest example in Section 8.1. In the most extreme case when $k=n$ is the complex dimension of $X$, this becomes the real Monge-Ampère equation

(2.67)

$$\det(W_{ij})=C.$$