Degenerate complex structures on surfaces and stability conditions

Abstract

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 $X^\vee$. On the other hand, recent work in the literature introduces a modular partial compactification, $\mathcal{A}\mathrm{Stab}(X)$, of a quotient of the space of stability conditions on $\mathcal{D}^{b}(X)$. We provide some mathematical evidence for the aforementioned heuristics by proving that for curves and surfaces, certain boundary strata in $\mathcal{A}\mathrm{Stab}(X)$ describe degenerate complex structures on $X^\vee$. Along the way, we introduce a criterion for producing boundary points of $\mathcal{A}\mathrm{Stab}(\mathcal{D})$ when $\mathcal{D}$ 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.