It is easy to see that an affine linear change in the coordinates $y_{j}$ (and hence an appropriate change in the coordinates $\check{x}_{j}$) results in a linear change of the coordinates $z_{j}$, so they induce a well-defined complex structure invariant under $\check{x}_{j}\mapsto\check{x}_{j}+1$, and hence a complex structure on $\check{X}(B)$. Then one computes that

$$\omega=\sum d\check{x}_{j}\wedge dy_{j}={i\over 2}\sum g^{jk}dz_{j}\wedge d\bar{z}_{k}$$

where $g_{ij}=\partial^{2}K/\partial y_{j}\partial y_{k}$. Then the metric is Ricci-flat if and only if $\det(g^{jk})=constant$, if and only if $\det(g_{jk})=constant$. ∎