Скачать или смотреть Daniel Halpern-Leistner, Cornell University: Dispatches from the ends of the stability manifold I

Daniel Halpern-Leistner, Cornell University: Dispatches from the ends of the stability manifold I

The manifold of Bridgeland stability conditions parameterizes a homological structure on a triangulated category that is analogous to a Kaehler structure on a projective variety. Recently, I have proposed a "noncommutative minimal model program" in which the quantum differential equation of a projective variety determines paths toward infinity in the stability manifold of that variety, and that these paths can be used to define canonical (semiorthogonal)decompositions of its derived category. It is natural to ask if these paths actually converge in some partial compactification of the stability manifold.

I will discuss a partial compactification of the stability manifold with a nice modular interpretation, the space of augmented stability conditions. To do this, I will introduce a structure on a triangulated category that we call a multi-scale decomposition, which generalizes a semiorthogonal decomposition, and a new moduli space of multi-scale lines that is closely related to the moduli spaces of multi-scale differentials which are of recent interest in dynamics. The main conjecture about the space of augmented stability conditions is that it is a manifold with corners (in a specific way that I will explain).

One consequence: If this conjecture holds for any smooth and proper dg-category, then any stability condition on a smooth and proper dg-category admits proper moduli spaces of semistable objects.