Let ${X}_{mer}=(X_{t})_{t\to 0}$ be a maximally degenerating algebraic Calabi-Yau manifold of dimension $n$ over ${{\bf C}}_{t}^{mer}$. We denote by $B$ the Gromov-Hausdorff limit of our family (see Conjecture 1, Section 5.1). Its connected oriented open dense part $B^{sm}$ carries a ${\bf Z}$-affine structure with the covariant lattice $T^{{\bf Z}}$.

Recall that according to the picture of collapse presented in Section 5.1 there is a canonical isotopy class of embeddings from a torus bundle $p:X_{t}^{\prime}\to B^{sm}$ to the complex manifold $X_{t}$ for all sufficiently small $t\neq 0$. Let us denote by $[\gamma_{0}]\in H_{n}(X_{t}^{\prime},{{\bf Z}})$ the fundamental class of the fiber of $p$. This is the homology class of a singular chain in $X_{t}^{\prime}$ which projects to a point by $p$.

Let $H_{n}^{\leq 1}(X_{t}^{\prime},{{\bf Z}})\subset H_{n}(X_{t}^{\prime},{{\bf Z}})$ be the subgroup generated by homology classes of chains which are projected into graphs in $B^{sm}$. It follows from the definition that we have an epimorphism

$$J_{a}:H_{1}(B^{sm},{\textstyle\bigwedge^{n-1}}T^{{\bf Z}})\twoheadrightarrow H_{n}^{\leq 1}(X_{t}^{\prime},{{\bf Z}})/{{\bf Z}}[\gamma_{0}]$$

similar to the homomorphims $J_{s}$ defined in the symplectic case (see Section 3.1.1). The following formula defines a homomorphism of groups

$$P:H_{n}^{\leq 1}(X_{t}^{\prime},{{\bf Z}})/{{\bf Z}}[\gamma_{0}]\to({{\bf C}}_{t}^{mer})^{\times},\,\,\,\,[\gamma]\mapsto\exp\left(2\pi i{\int_{[\gamma]}\Omega_{t}\over\int_{[\gamma_{0}]}\Omega_{t}}\right)\,\,.$$

We will call $P$ the period map. Notice that ${{\bf Z}}[\gamma_{0}]:=H_{n}^{\leq 0}(X_{t}^{\prime},{{\bf Z}})\subset H_{n}^{\leq 1}(X_{t}^{\prime},{{\bf Z}})$ is a low degree part of the limiting Hodge filtration on the homology of Calabi-Yau manifold $X_{t}$. Non-zero complex numbers

$$\exp\left(2\pi i{\int_{[\gamma_{i}]}\Omega_{t}\over\int_{[\gamma_{0}]}\Omega_{t}}\right)\,\,,$$

where $\gamma_{i}$ is a set of generators of $H_{n}^{\leq 1}(X_{t}^{\prime},{{\bf Z}})/H_{n}^{\leq 0}(X_{t}^{\prime},{{\bf Z}})$ are called flat coordinates in Mirror Symmetry (see e.g. [Mor]). Those are local coordinates near a point close to the “cusp” of the moduli space of complex structures (local Torelli theorem).

The orientation of $B^{sm}$ gives rise to an isomorphism $\bigwedge^{n-1}T^{{\bf Z}}\simeq(T^{\ast})^{{\bf Z}}$. Therefore, combining maps $J_{a},P$ and the above isomorphism we obtain a homomorphism

$$\widetilde{P}:H^{1}(B^{sm},(T^{\ast})^{{\bf Z}})\to\left({{\bf C}}_{t}^{mer}\right)^{\times}\,\,.$$