Let $ M$ be a $2n$-dimensional compact and connected manifold.
Suppose there is $\Omega\in\Omega^{1}(M,\mathbb{C}) $ a closed complex form whose real and imaginary parts represent linearly independent rational cohomology classes and such that
$$ \omega^{n-1} \wedge \Omega \wedge \bar{\Omega}\neq 0$$ (Here $\omega$ is a symplectic form.)
Define $\Lambda=[\Omega]H\_{1}(M,\mathbb{Z}) $. We define a map $\pi:M\longrightarrow \mathbb{C}/\Lambda $ by 
$$ \pi(p)=\int\_{p\_{0}}^{p}\Omega. $$


1) Why is$\Lambda$ a cocompact lattice?


2) Why is $\pi$ a surjective submersion?



Submersion to $ T^{2}$
1. Note first that $\Lambda\subset \mathbb C$ is a subgroup generated by a finite collection of numbers (periods) of the type $\alpha+i\beta$, where $\alpha,\, \beta\in \mathbb Q$. Such a subgroup is clearly discreet so it is either $\mathbb Z$ or $\mathbb Z^2$. The only possibility for such subgroup to be isomorphic to $\mathbb Z$ is when all such periods are proportional to each other (over $\mathbb Q$). However such periods can not be all proportional since otherwise the cohomology classes of $Re(\Omega)$ and $Im(\Omega)$ will be proportional. Hence the group of periods is $\mathbb Z^2$ and it is a lattice in $\mathbb C$.This proves the first assertion.
2. Assume by contradiction that the map $\pi$ is not surjective. In this case the pullback map $\pi^\*: H^2(\mathbb C/\Lambda,\mathbb C)\to H^2(M, \mathbb C)$ is trivial. However by construction the class of $\Omega\wedge \overline \Omega$ belongs to the image of the latter map. This contradicts $\omega^{n-1}\wedge \Omega\wedge \overline \Omega\ne 0$.
3. Note that the $2n$-form $\omega^{n-1}\wedge\Omega\wedge \overline \Omega$ is vanishing at every point where the differential of $\pi$ has rank less than $2$. So in case $\omega^{n-1}\wedge\Omega\wedge \overline \Omega$ is a non-vanishing volume form the map $\pi$ is submersive.


PS. Note that if case we only ask $\omega^{n-1}\wedge\Omega\wedge \overline \Omega$ to be non-zero in cohomology, $\pi$ need not be a submertion, it is easy construct such examples on $T^2$ - just take $\Omega$ that vanishes on a small open disk in $T^2$.


