Question

The Novikov cohomology is the cohomology of the differential $d_{\theta}$:

$$d_{\theta}\omega = d\omega + \theta \wedge \omega$$

$$d\theta=0$$

$$\Delta_{\theta}= d_{\theta} d_{\theta}^* + d_{\theta}^* d_{\theta}$$

Have we a Hodge decomposition for a closed form $\omega$:

$$\omega = \omega_0 + d_{\theta} \alpha$$

with $\Delta_{\theta} \omega_0=0$?