# Can there be a nonzero period integral of this form?

I have been trying to compute the following integral:

$$\displaystyle \int_{\gamma} \frac{g \Omega_{\mathbb{P}^3}}{f^2}$$

where:

$\gamma$ is the torus cycle $\{ |z_1| =|z_2| =|z_3| =|z_4| =1 \}$,

$g \in V_1 = \Gamma(\mathbb{P}^3, O(2))$,

$f \in V_2 = \Gamma(\mathbb{P}^3, O(3))$, and

$\Omega_{\mathbb{P}^3} = z_1dz_2dz_3dz_4 - z_2dz_1dz_3dz_4 + z_3dz_1dz_2dz_4 - z_4dz_1dz_2dz_3$.

I have been trying to find particular sections $f$ and $g$ such that when I evaluate $g$ at a point, I obtain a nonzero function with domain space $V_2^{*}$.

Any suggestions are welcome!