Categorical realization of collapsing subsurfaces and perverse schobers

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-M\"oller-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.