Vanishing of Local Cohomology with Applications to Hodge Theory

Let $\mathbf{\text{H}}=((H,F^{\bullet }),L)$ be a polarized variation of Hodge structure on a smooth quasi-projective variety $U\mathrm{.}$ By M. Saito’s theory of mixed Hodge modules, the variation of Hodge structure $\mathbf{\text{H}}$ can be viewed as a polarized Hodge module $\mathcal{M}\in HM(U)\mathrm{.}$ Let $X$ be a compactification of $U,$ and $j:U\hookrightarrow X$ is the natural map. In this paper, we use local cohomology with mixed Hodge module theory to study $j_{+}\mathcal{M}\in D^{b}MHM(X)\mathrm{.}$ In particular, we study the graded pieces of the de Rham complex $G{r}_p^{F}DR(j_{+}\mathcal{M})\in D_{coh}^b(X),$ and the Hodge structure of $H^i(U,L)$ for $i$ in sufficiently low degrees.