Definition 2.5 Let $m\geqslant 2$. A Calabi–Yau $m$-fold, or CY $m$-fold for short, is a quadruple $(X,J,\omega,\Omega)$ such that $(X,J)$ is a compact $m$-dimensional complex manifold, $\omega$ the Kähler form of a Kähler metric $g$ on $X$, and $\Omega$ a non-vanishing holomorphic $(m,0)$-form on $X$ which satisfies

$$\omega^{m}/m!=(-1)^{m(m-1)/2}(i/2)^{m}\Omega\wedge\bar{\Omega}.$$

(7)

Then for each $x\in X$ there exists an isomorphism $T_{x}X\cong\mathbin{\mathbb{C}}^{m}$ that identifies $g_{x},\omega_{x}$ and $\Omega_{x}$ with the flat versions $g,\omega,\Omega$ on $\mathbin{\mathbb{C}}^{m}$ in (1).