Categorical realization of collapsing subsurfaces and perverse schobers
Li Fan, Suiqi Lu
TL;DR
The paper develops a categorification framework for collapsed surfaces equipped with quadratic differentials by employing Verdier quotients and quotient perverse schobers. It proves an isomorphism between the principal parts of exchange graphs of hearts in the Verdier quotient and those of mixed-angulations on the collapsed surface, extending prior work to include higher-order poles. A dual, schober-based construction yields a short exact sequence of triangulated categories, tying collapsing to semiorthogonal decompositions and enabling an isomorphism between moduli of framed quadratic differentials and stability spaces of the quotient category. The results give a Bridgeland–Smith type correspondence in this generalized setting and offer new perspectives on compactifications of stability-condition spaces via degenerations of quadratic differentials. Overall, the work provides both a combinatorial and a categorical path to understand wall-crossing, stability conditions, and the geometry of moduli spaces in the presence of arbitrary zeros and poles.
Abstract
We study the categorification of collapsed Riemann surfaces with quadratic differentials allowing arbitrary order zeros and poles via the Verdier quotient. We establish an isomorphism between the exchange graph of hearts in the quotient category and the exchange graph of mixed-angulations on the collapsed surface. This extends the work of Barbieri-Möller-Qiu-So, who studied Verdier quotients of 3-Calabi-Yau categories and collapsed surfaces without simple poles. We use two methods: a combinatorial approach, and another based on the global sections of a quotient perverse schober. As an application, we describe the Bridgeland stability conditions in terms of quadratic differentials on the collapsed surface.