Weight filtration and generating level

We study the canonical mixed Hodge module structure associated to the ${\mathcal{D}}_X$-module $\mathcal{M}(f^{-\alpha }):={\mathcal{O}}_X(\ast f)f^{-\alpha }$. We particularly focus on the weight filtration and extend many known results to the weighted setting. We obtain new relations between Hodge theory and birational geometry. We derive a general formula for the Hodge and weight filtrations on $\mathcal{M}(f^{-\alpha })$, and use this to obtain results concerning the largest weight of $\mathcal{M}(f^{-\alpha })$ and the generating level of weight filtration steps. Finally, we obtain expressions for several classes of divisor, including certain parametrically prime divisors.