In this section we discuss semi-flat forms and metrics, extending some results in [18, 20] to our setting.

In general a closed real $(1,1)$-form $\omega_{SF}$ on an open set $U\subset M\backslash S$ will be called semi-flat if its restriction to each torus fiber $M_{y}\cap U$ with $y\in f(U)$ is a flat metric, which we will always assume to be cohomologous to $\omega_{M}|_{M_{y}}$. If $\omega_{SF}$ is also Kähler then we will call it a semi-flat metric. Semi-flat forms can also be defined when the fibers $M_{y}$ are not tori but general Calabi-Yau manifolds, by requiring that the restriction to each fiber be Ricci–flat (see [34, 38]). They were first introduced by Greene-Shapere-Vafa-Yau in [14].

Fix now a small ball $B\subset N\backslash f(S)$ with coordinates $y=(y_{1},\dots y_{m})$, and consider the preimage $f:U=f^{-1}(B)\to B$. This is a holomorphic family of complex tori, and if $B$ is small enough it has a holomorphic section $\sigma_{0}$, which we also fix. We can then define a complex Lie group structure on each fiber $M_{y}=f^{-1}(y)$ with unit $\sigma_{0}(y)$. We claim that this family is locally isomorphic to a family of the form $f^{\prime}:(B\times\mathbb{C}^{n-m})/\Lambda\to B,$ where $h:\Lambda\to B$ is a lattice bundle with fiber $h^{-1}(y)=\Lambda_{y}\cong\mathbb{Z}^{2n-2m}$, so that $M_{y}\cong\mathbb{C}^{n-m}/\Lambda_{y}.$ To see this, note that each fiber $M_{y}=f^{-1}(y)$ is a torus biholomorphic to $\mathbb{C}^{n-m}/\Lambda_{y}$ for some lattice $\Lambda_{y}$ that varies holomorphically in $y$. We choose a basis $v_{1}(y),\dots,v_{2n-2m}(y)$ of this lattice, which varies holomorphically in $y$. Given these lattices we can construct the family $f^{\prime}$ by taking the quotient of $B\times\mathbb{C}^{n-m}$ by the $\mathbb{Z}^{2n-2m}$-action given by $(n_{1},\dots,n_{2n-2m})\cdot(y,z)=(y,z+\sum_{i}n_{i}v_{i}(y))$, where $z=(z_{1},\dots,z_{n-m})\in\mathbb{C}^{n-m}$. Note that different choices of generators give isomorphic quotients. By construction the fiber $f^{\prime-1}(y)$ is biholomorphic to $f^{-1}(y)$ for all $y\in B$. A theorem of Kodaira-Spencer [23] (see also [43, Satz 3.6]) then implies that the families $f$ and $f^{\prime}$ are locally isomorphic, so up to shrinking $B$ there exists a biholomorphism $(B\times\mathbb{C}^{n-m})/\Lambda\to U$ compatible with the projections to $B$, proving our claim. With this identification, the section $\sigma_{0}:B\to U$ is induced by the map $B\to B\times\mathbb{C}^{n-m}$ given by $y\mapsto(y,0)$.

Composing this biholomorphism with the quotient map $B\times\mathbb{C}^{n-m}\to(B\times\mathbb{C}^{n-m})/\Lambda$ by the $\mathbb{Z}^{2n-2m}$-action we get a holomorphic map $p:B\times\mathbb{C}^{n-m}\to U$ such that $f\circ p(y,z)=y$ for all $(y,z)$, and $p$ is a local isomorphism (the map $p$ is also the universal covering map of $U$).

We now assume that $M$ is projective and $[\omega_{M}]$ is an integral class, so each complex torus fiber $M_{y}$, $y\in B$, can be polarized by $[\omega_{M}]$, which gives an ample polarization of type $(d_{1},\ldots,d_{n-m})$ for some sequence of integers $d_{1}|d_{2}|\cdots|d_{n-m}$. By [3], Proposition 8.1.1, one can then assume that $\Lambda$ is generated by $d_{1}e_{1},\ldots,d_{n-m}e_{n-m},Z_{1},\ldots,Z_{n-m}\in\mathbb{C}^{n-m}$, where $e_{1},\ldots,e_{n-m}$ is the standard basis for $\mathbb{C}^{n-m}$. Furthermore, the matrix $Z$ with columns $Z_{1},\ldots,Z_{n-m}$ must satisfy $Z=Z^{t}$ and ${\rm Im}Z$ positive definite. Also, on the fibre $M_{y}$, the Kähler form $\sum_{i,j}\sqrt{-1}(\operatorname{Im}Z)_{ij}dz^{i}\wedge d\bar{z}^{j}$ is cohomologous to $\omega_{M}|_{M_{y}}$. Let

$$g_{ij}=({\rm Im}Z)^{-1}_{ij}.$$

Note that $Z$ depends on $y\in B$, as does $g_{ij}$. Recall that we have the fiber coordinates $z_{1},\dots,z_{n-m}$. Consider the function

$$\eta(y,z)=\sum_{i,j}-{\frac{g_{ij}(y)}{2}}\left((z_{i}-\bar{z}_{i})(z_{j}-\bar{z}_{j})\right).$$

We would first like to show that $\sqrt{-1}\partial\bar{\partial}\eta$ is invariant under translation by flat sections of the Gauss-Manin connection on $B\times\mathbb{C}^{n-n}$ (this is the connection on this bundle such that sections of $\Lambda$ are flat sections of the bundle). It is enough to check invariance under translation by $\lambda s$ for $s$ one of the generators of $\Lambda$, $\lambda\in\mathbb{R}$. First, consider the composition of $\eta$ with a general translation $z_{i}\mapsto z_{i}+\tau_{i}(y)$:

		$\displaystyle\sum_{i,j}-{\frac{g_{ij}}{2}}\left((z_{i}+\tau_{i}-\bar{z}_{i}-\bar{\tau}_{i})(z_{j}+\tau_{j}-\bar{z}_{j}-\bar{\tau}_{j})\right)$
	$\displaystyle={}$	$\displaystyle\eta-\sum_{i,j}{\frac{g_{ij}}{2}}\left((\tau_{i}-\bar{\tau}_{i})(z_{j}-\bar{z}_{j})+(\tau_{j}-\bar{\tau}_{j})(z_{i}-\bar{z}_{i})+(\tau_{i}-\bar{\tau}_{i})(\tau_{j}-\bar{\tau}_{j})\right)$
	$\displaystyle={}$	$\displaystyle\eta-\sum_{i,j}g_{ij}\left((\tau_{i}-\bar{\tau}_{i})(z_{j}-\bar{z}_{j})+{\frac{1}{2}}(\tau_{i}-\bar{\tau}_{i})(\tau_{j}-\bar{\tau}_{j})\right),$

the last equality by the symmetry $g_{ij}=g_{ji}$. We now consider two cases. If $\tau_{i}=\lambda\delta_{ik}$ for some $k$, so that $\tau_{i}$ is real, then in fact the above formula reduces to $\eta$, so $\eta$ is itself invariant under this translation. Secondly, if we take $\tau_{i}=\lambda Z_{ik}$ for some $k$, $\lambda\in\mathbb{R}$, we obtain

		$\displaystyle\eta-\sum_{i,j}({\rm Im}Z)^{-1}_{ij}\left(2\lambda\sqrt{-1}({\rm Im}Z)_{ik}(z_{j}-\bar{z}_{j})-2\lambda^{2}({\rm Im}Z)_{ik}({\rm Im}Z)_{jk}\right)$
	$\displaystyle={}$	$\displaystyle\eta-\sum_{j}2\delta_{jk}\lambda\sqrt{-1}(z_{j}-\bar{z}_{j})-2\lambda^{2}\delta_{jk}({\rm Im}Z)_{jk}.$

Applying $\partial\bar{\partial}$ kills the correction term, so $\sqrt{-1}\partial\bar{\partial}\eta$ is invariant under this action. This means that $\sqrt{-1}\partial\overline{\partial}\eta$ is the pullback under $p$ of a two-form $\omega_{SF}$ on $U$

(3.1)

$$p^{*}\omega_{SF}=\sqrt{-1}\partial\overline{\partial}\eta,$$

and $\omega_{SF}$ is semi-flat since its restriction to a fiber is $\sqrt{-1}\sum_{i,j}g_{ij}(y)dz^{i}\wedge d\bar{z}^{j}$, a flat metric on $M_{y}$ cohomologous to $\omega_{M}|_{M_{y}}$. Note that the function $\eta$ on $B\times\mathbb{C}^{n-m}$ has the scaling property

(3.2)

$$\eta(y,\lambda z)=\lambda^{2}\eta(y,z),$$

for all $\lambda\in\mathbb{R}$.

We now claim that on $U$ the semi-flat form $\omega_{SF}$ is nonnegative definite. To check this, it is enough to check at one point on each fiber, because of the invariance of this form. We check at the point $z_{1}=\cdots=z_{n-m}=0$, where the form is $\sqrt{-1}\sum_{i,j}g_{ij}dz^{i}\wedge d\bar{z}^{j}$, which is clearly nonnegative definite. It follows that $\omega_{SF}\geqslant 0$, and moreover that given any Kähler metric $\omega^{\prime}$ on $B$ the form $\omega_{SF}+f^{*}\omega^{\prime}$ is a semi-flat Kähler metric on $U$.

Suppose now that we have a holomorphic section $\sigma:B\to U$ of the map $f$. We will denote by $T_{\sigma}:U\to U$ the fiberwise translation by $\sigma$ (with respect to the section $\sigma_{0}$). If we choose any local lift of $\sigma$ to $B\times\mathbb{C}^{n-m}$, given by $y\mapsto(y,\tilde{\sigma}(y))$, then the translation $T_{\sigma}$ is induced by the map $B\times\mathbb{C}^{n-m}\to B\times\mathbb{C}^{n-m}$ given by $(y,z)\mapsto(y,z+\tilde{\sigma}(y))$ (the choice of lift $\tilde{\sigma}$ is irrelevant). We also have a map $T_{-\sigma}:U\to U$ given by fiberwise translation by $-\sigma$ (with respect to $\sigma_{0}$), which is induced by $(y,z)\mapsto(y,z-\tilde{\sigma}(y))$. The two translation are biholomorphisms of $U$ and are inverses to each other. For later purposes, we will need the following version of the $\partial\overline{\partial}$-Lemma, which is analogous to [18, Lemma 4.3] (see also [20, Proposition 4.6]), except that we work away from the singular fibers.