Semi-stable degenerations of Calabi-Yau manifolds and mirror P=W conjecture

Abstract: Mirror symmetry for a semi-stable degeneration of a Calabi-Yau manifold was first investigated by Doran-Harder-Thompson when the degeneration fiber is a union of two (quasi)-Fano manifolds. They propose a topological construction of a mirror Calabi-Yau that is a gluing of two Landau-Ginzburg models mirror to those Fano manifolds. We extend this construction to a general type semi-stable degeneration. As each component in the degeneration fiber comes with the simple normal crossing anti-canonical divisor, one needs the notion of a hybrid Landau-Ginzburg model, a multi-potential analogue of classical Landau-Ginzburg models. We show that these hybrid LG models can be glued to provide a topological mirror candidate of the Calabi-Yau which is also equipped with the fibration over $\mathbb{P}^N$. Furthermore, it is predicted that the perverse Leray filtration associated to this fibration is mirror to the monodromy weight filtration on the degeneration side. We explain how this can be deduced from the original mirror P=W conjecture.