Degenerate complex structures on surfaces and stability conditions

Physical heuristics suggest that the space of Bridgeland stability conditions of a projective Calabi-Yau manifold X should be related to the moduli space of complex structures on its mirror X^. On the other hand, recent work in the literature introduces a modular partial compactification, AStab(X), of a quotient of the space of stability conditions on Dᵇ(X). We provide some mathematical evidence for the aforementioned heuristics by proving that for curves and surfaces, certain boundary strata in AStab(X) describe degenerate complex structures on X^. Along the way, we introduce a criterion for producing boundary points of AStab(D) when D is an arbitrary k-linear triangulated category. As a byproduct, we deduce the existence of Bridgeland stability conditions on certain quotients of the derived categories of K3 and Enriques surfaces.