  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.

Algebraic Geometry