Asymptotic and polynomial stabilities in Bridgeland stability conditions

Abstract

In algebraic geometry, moduli spaces are parameter spaces of geometric objects. Moduli spaces ofsheaves, in particular, have been an important area of study over the past few decades, partly because of their tight connections with theoretical physics. A typical way to construct a moduli space is to begin with a criterion that decides which sheaves will be parametrized by the moduli space; such a criterion is called a stability condition. Since different choices of stability conditions give rise to different modulispaces, whose differing properties play fundamental roles in different contexts, it is important to understand the relations among different types of stability conditions. In this research project, we have two main goals: (1) To understand how moduli spaces behave under symmetries called autoequivalenes in algebraic geometry, and how the resulting behaviours are related to the S-duality conjecture in physics. (2) To understand the connections among Gieseker stability (a classical stability condition), Bridgeland stability conditions, polynomial stability conditions, and the newly defined notion of asymptotic stability conditions.Towards goal (1), the starting point will be the ongoing joint work between Lo and Martinez, andprevious work by Lo. These works study the effects of autoequivalences on Bridgeland stability conditions, particularly on elliptic fibrations. This project will begin the process of refining these results, so new tools can be created to understand the interactions among autoequivalences, Bridgeland stability conditions, and S-duality. Towards goal (2), the starting point will be the recent joint work of Jardim and Maciocia, as well as their joint work with Martinez. These works study the notion of asymptotic stability introduced by Jardim andMaciocia, and appear to be related to previous works on polynomial stability conditions. This project will begin the process of understanding the connections between the new notion of asymptotic stability and existing notions of Bridgeland and polynomial stability conditions, both on the level of stability conditions and on the level of moduli spaces. (AU)