Geometry and Analysis Seminar

We introduce a notion of refined Harder-Narasimhan filtration, defined abstractly for algebraic stacks satisfying natural conditions. Examples include moduli stacks of objects at the heart of a Bridgeland stability condition, moduli stacks of K-semistable Fano varieties, moduli of principal bundles on a curve, and quotient stacks. We will explain how refined Harder-Narasimhan filtrations are closely related both to stratifications and to the asymptotics of certain analytic flows, relating and expanding work of Kirwan and Haiden-Katzarkov-Kontsevich-Pandit, respectively. In the case of quotient stacks by the action of a torus, the refined Harder-Narasimhan filtration can be computed in terms of convex geometry.

Please note unusual day and room. 

Motivated by the SYZ conjecture, it is expected that $G_2$ and Spin(7)-manifolds also admit calibrated fibrations. One potential way to construct examples is via gluing of complex fibrations, as in the program of Kovalev. For this to succeed we need that the fibration property is stable under deformation of the ambient Spin(7)-structure. Here the main difficulty lies in the analysis of the singular fibres. In this talk I will present a stability result for fibrations with conically singular Cayleys modeled on the complex cone $\{x^2 + y^2 + z^2 = 0\}$ in ${\mathbb C}^3$.