Hodge Theory and Grobner Degenerations

In this paper, we investigate the behavior of mixed Hodge structures under square-free Grobner degenerations. We show that the weight-zero part of singular cohomology and the zeroth graded piece of the Hodge filtration are preserved under such degenerations, complementing the homological rigidity established in the seminal work of Conca and Varbaro. We use these results to settle their conjecture concerning the singularities of homogeneous coordinate rings admitting such degenerations.