Let $V^{ij}$, respectively $W^{pq}$, be real symmetric, respectively hermitian, positive definite matrices of smooth functions on $B^{\circ}$, locally given by some potential function $\Phi$:

(1)

$$V^{ij}=\frac{\partial^{2}\Phi}{\partial u_{j}\partial u_{j}},\quad W^{pq}=-4\frac{\partial^{2}\Phi}{\partial\eta_{p}\partial\bar{\eta}_{q}},\qquad 1\leq i,j\leq n\quad n+1\leq p,q\leq N.$$

Then the following $\mathfrak{t}$-valued 2-form is closed:

(2)

$$F_{j}=\sqrt{-1}\left(\frac{1}{2}\frac{\partial W^{pq}}{\partial u_{j}}d\eta_{p}\wedge d\bar{\eta}_{q}+\frac{\partial V^{ij}}{\partial\eta_{p}}du_{i}\wedge d\eta_{p}-\frac{\partial V^{ij}}{\partial\bar{\eta}_{q}}du_{i}\wedge d\bar{\eta}_{q}\right).$$

Suppose, in addition, that $\det V^{ij}=\det W^{pq}$ and $(F_{1},\dots,F_{n})$ is in the cohomology class $2\pi[\nu]$. Then there exist a connection on the bundle $M\to B$ with associated 1-forms $A_{i}$ and the curvature $(F_{1},\dots,F_{n})$ such that $M$ is a Kähler manifold with Ricci-flat metric given by

(3)

$$h=(V^{-1})^{ij}dz_{i}\otimes d\bar{z}_{j}+W^{pq}d\eta_{p}\otimes d\bar{\eta}_{q},$$

where $dz_{j}=V^{ij}du_{i}+\sqrt{-1}\cdot A_{j}$ and $d\eta_{p}$ form a basis of holomorphic 1-forms. The holomorphic $N$-form and the Kähler form:

(4)

$$\Omega=\wedge_{j=1}^{k}dz_{j}\bigwedge\wedge_{p=1}^{l}d\eta_{p},\qquad\omega=du_{j}\wedge A_{j}+\frac{\sqrt{-1}}{2}\cdot W^{pq}d\eta_{p}\wedge d\bar{\eta}_{q}$$

are compatible in the sense that $\Omega\wedge\bar{\Omega}=const\cdot\omega^{N}$.