Geometry and Physics Seminar: Daniel Halpern-Leistner

In order to construct moduli spaces in algebraic geometry, one typically must specify a notion of semi-stability for the objects one wishes to parameterize. To the objects that are omitted, the unstable objects, one can often associate a real number which measures "how unstable" that object is. In fact we can think of the moduli stack of all objects as stratified by locally closed substacks corresponding to objects of varying degrees of instability. The key examples of this phenomenon are the Kempf-Ness stratification of the unstable locus in GIT and the Shatz stratification of the moduli of G-bundles on a smooth projective curve. I will discuss a framework for describing stability conditions and stratifications of an arbitrary algebraic stack which provide a common generalization of these examples. Time permitting, I will discuss how some commonly studied moduli problems, such as the moduli of K-stable varieties and the moduli of Bridgeland-semistable complexes on a smooth projective variety, fit into this framework. One key construction assigns to any point in an algebraic stack a potentially large topological space parameterizing all possible `iso-trivial degenerations' of that point. When the stack is BG for a reductive G, this recovers the spherical building of G, and when the stack is X/T for a toric variety X, this recovers the support of the fan of X.