(Theorem 1.5, sketch) By formula (1.6) and the integrability condition (1.10),

(1.12)

$$dF_{j}=\frac{\sqrt{-1}}{2}\left(\frac{\partial^{2}W^{p\bar{q}}}{\partial\mu_{i}\partial\mu_{j}}+4\frac{\partial^{2}V^{ij}}{\partial\eta_{p}\partial\bar{\eta}_{q}}\right)d\mu_{i}\wedge d\eta_{p}\wedge d\bar{\eta}_{q},$$

so the closedness of $F_{j}$ is equivalent to (1.11). Since $\frac{1}{2\pi}F$ represents the appropriate first Chern class, $F$ must be the curvature of a $T^{\mathfrak{n}}$-connection $\vartheta$. Modulo gauge $\vartheta$ admits the local formula

(1.13)

$$\vartheta_{j}=\sqrt{-1}\{\frac{\partial^{2}\Phi}{\partial\eta_{p}\partial\mu}d\eta_{p}-\frac{\partial^{2}\Phi}{\partial\bar{\eta}_{p}\partial\mu}d\bar{\eta}_{p}\}.$$

Gauge equivalent choices of the connection define the structures on $M$ up to holomorphic isometry.

The integrability of the complex structure follows from the fact that the differential ideal generated by $(1,0)$ forms is closed:

(1.14)

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

using (1.10)(1.6) and the definition of $\zeta_{j}$. The Kähler condition $d\omega=0$ follows from (1.10)(1.6). The Calabi-Yau condition follows from the more general formula

$$\omega^{N}=\det(W^{p\bar{q}})\det(V^{ij})^{-1}\frac{N!}{2^{N}}\sqrt{-1}^{N^{2}}\Omega\wedge\overline{\Omega}.$$

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