Semistable degenerations of Calabi-Yau manifolds and mirror P=W conjectures

Mirror symmetry for a semistable degeneration of a Calabi-Yau manifold was first investigated by Doran-Harder-Thompson when the degenerate fiber is a union of two quasi-Fano manifolds. They proposed a topological construction of a mirror Calabi-Yau by gluing of two Landau-Ginzburg models that are mirror to those Fano manifolds. We extend this construction to a general type semistable degeneration where the dual boundary complex of the degenerate fiber is the standard N-simplex. Since each component in the degenerate fiber comes with the simple normal crossing anticanonical divisor, one needs the notion of a hybrid Landau-Ginzburg model - a multipotential analogue of classical Landau-Ginzburg models. We show that these hybrid Landau-Ginzburg models can be glued to be a topological mirror candidate for the nearby Calabi-Yau, which also exhibits the structure of a Calabi-Yau fibration over P-N. Furthermore, it is predicted that the perverse Leray filtration associated to this fibration is mirror to the monodromy weight filtration on the degeneration side [12]. We explain how this can be deduced from the original mirror P=W conjecture [18].