Limits of Hodge structures via holonomic D-modules

We construct the limiting mixed Hodge structure of a degeneration of compact K\"ahler manifolds over the unit disk with a possibly non-reduced simple normal crossing singular central fiber via holonomic $\mathscr D$-modules, generalizing some results of Steenbrink. Our limiting mixed Hodge structure does not carry a $\mathbb Q$-structure; instead, we use sesquilinear pairings on $\mathscr D$-modules as a replacement. The limiting mixed Hodge structure can be computed by the cohomology of the cyclic coverings of certain intersections of components of the central fiber. Additionally, we prove the local invariant cycle theorem in this setting.