(cf. Theorem 2.1 in [31]) Let $V^{ij}$, respectively $W^{p\bar{q}}$, be real symmetric positive definite/Hermitian matrices of smooth functions on $\mathcal{B}^{0}$, locally given by some potential function $\Phi$:

(1.5)

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

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

(1.6)

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

Suppose further that $\frac{1}{2\pi}(F_{1},\ldots,F_{\mathfrak{n}})$ is in the cohomology class $c_{1}\in H^{2}(\mathcal{B}^{0},\mathfrak{t}_{\mathbb{Z}})$. Then there exists a connection $\vartheta$ on the principal bundle $M\to\mathcal{B}^{0}$ with curvature $d\vartheta_{i}=F_{i}$ for $i=1,\ldots,\mathfrak{n}$, such that $M$ is a Kähler manifold with metric tensor

(1.7)

$$h=(V^{-1})^{ij}\zeta_{i}\otimes\bar{\zeta}_{j}+W^{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q},\quad\omega=d\mu_{j}\wedge\vartheta_{j}+\frac{\sqrt{-1}}{2}W^{p\bar{q}}d\eta_{p}\wedge d\bar{\eta}_{q},$$

where $\zeta_{j}=V^{ij}d\mu_{i}+\sqrt{-1}\vartheta_{j}$ and $\eta_{p}$ form a basis of type (1,0) forms which defines an integrable complex structure. There is a nowhere vanishing holomorphic form on $M$:

(1.8)

$$\Omega=\wedge_{j=1}^{\mathfrak{n}}(-\sqrt{-1}\zeta_{j})\bigwedge\wedge_{p=1}^{N-\mathfrak{n}}d\eta_{p}.$$

The Calabi-Yau condition $\omega^{N}=\frac{N!}{2^{N}}\sqrt{-1}^{N^{2}}\Omega\wedge\overline{\Omega}$ is equivalent to the equation

(1.9)

$$\det(V^{ij})=\det(W^{p\bar{q}}).$$