dipendence of algebraic funtions Let $S$ a complex compact algebraic surface. Write $K(S)$ instead of the field of rational functions on $S$.
Suppose that there exist two holomorfic 1 forms $\omega_1$,$\omega_2$ in $H^0(S,\Omega^1_S)$ such that the cup product of them is zero, i.e. $\omega_1 \wedge \omega_2=0$.
Using the previous equality we can say that there exist a function $g \in K(S)$ such that $\omega_2=g\omega_1$. Then, using Stokes's theorem, we get $d\omega_1=d\omega_2=0$ that implies $\omega_1 \wedge dg=0$ and so $\omega_1=fdg$ for some rational functions on $S$.
Now $d\omega_1=0$ implies $df \wedge dg=0$.
Is it true that $f$ and $g$ are algebrically dipendent?
 A: The answer is yes. To show this we will construct a map from surface $S$ to a curve $E$ such that all forms and functions in the questions are pull-backs of forms and functions on curve $E$. Since transcendence degree of $K(E)$ is one, the functions will be dependent. 
How to construct $E$? The main reference is [Phillip Griffiths, Joseph Harris: Principles of Algebraic Geometry, Ch. IV, 5.2] Integrating $\omega_1$ and $\omega_2$ (locally) on $\widetilde U\subset S$ we get a map
$$
\pi: \widetilde U\to U\subset\mathbb C^2.
$$
Actually, the image $\pi(\widetilde U)=Z$ is a curve and for every point $x\in Z$  the restrictions 
$$
\omega_1|_{\pi^{-1}(x)}\ \mbox{and}\ \omega_2|_{\pi^{-1}(x)}
$$
are trivial. After a suitable choice of coordinate on $Z$ one can find that the map $\pi:\widetilde U\to Z$ is given by $g=\frac{\omega_1}{\omega_2}$ (to check this, one need to consider jacobian of the map $\pi:\widetilde U\to U$). 
Function $g=\frac{\omega_1}{\omega_2}$ is holomorphic and defines extended map $\pi:S\to\mathbb P^1$. For a general point $\lambda\in \mathbb P^1$ the pre-image $\pi^{-1}(\lambda)$ is a union of irreducible divisors $C_{\lambda,i}$ (which do not intersect because of the smoothness of $\pi^{-1}(\lambda)$ for general $\lambda$). Also, restrictions of $\omega_1$ and $\omega_2$ on each fiber are trivial. The set of divisors 
$$
E=\bigcup_{i,\lambda} \{C_{\lambda,i}\}
$$
is actually a (ramified) finite cover of $\mathbb P^1$ and is an algebraic curve (one can give more explicit construction, see [Phillip Griffiths, Joseph Harris: Principles of Algebraic Geometry]). We have a decomposition of $\pi$:
$$
S\to E\to \mathbb P^1,
$$
$$
p\mapsto (\lambda, i) \mapsto \lambda,
$$
where $(\lambda, i)$ is defined by $p\in C_{\lambda, i}$ (curves do not intersect).
Since $\omega_1$ and $\omega_2$ are constant along divisors $C_{\lambda,i}$, which are irreducible, they are pull-backs of forms on $E$, the same is true for $f$ and $g$ as well.
