Canonical polynomial filtrations in moduli theory

Andres Ibanez Nunez (University of Oxford)

Tuesday 10th October, 2023 16:00-17:00 110, Mathematics & Statistics Building

Abstract

We present a canonical stratification defined for general algebraic stacks admitting a good moduli space. The strata live in a stack of “polynomial filtrations”, and thus the stratification defines one canonical such filtration, the iterated balanced filtration, for every point of the stack. Examples include stacks of (1) Bridgeland semistable objects, (2) K-semistable Fano varieties and (3) Geometric Invariant Theory quotients, where the stratification was originally defined by Kirwan.

 

In the case of Bridgeland semistable objects, the iterated balanced filtration coincides with a filtration defined, using completely different methods, by Haiden-Katzarkov-Kontsevich-Pandit (HKKP) for artinian lattices. The HKKP filtration was used by the authors to describe the asymptotics of certain natural gradient flows on the space of metrics of quiver representations. We will illustrate our belief that, more generally, the iterated balanced filtration describes the asymptotics of the gradient flows appearing naturally in many other moduli problems.

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