Cambridge Complex Geometry Afternoon

The purpose of this talk is to investigate the theory of Z-stability and Z-critical metrics developed by Dervan, McCarthy and Sektnan. Its aim is to provide a general framework relating stability conditions to existence of solutions to geometric PDEs, in a way analogous to the YTD conjecture or the Kobayashi-Hitchin correspondence. In the special case of K-stability and cscK metrics, Legendre successfully applied equivariant localization, offering a new insight on some invariants involved. I will start by introducing the necessary tools in equivariant cohomology, then show that this method can be applied more broadly to the family of Z-stability conditions, and existence of Z-critical metrics.

Given any toric vector bundle, we may construct its parliament of polytopes. This is a generalization of the Newton polytope (or moment polytope) of a line bundle. This object contains a huge amount of information about the original bundle: notably on its global sections and its positivity. We can also easily know if the toric bundle is (semi-/poly-)stable with respect to any polarisation. I will give a combinatorial visualisation of stability of toric vector bundles.

An important category of geometric objects in algebraic geometry is smooth Fano varieties. These have been classified in 1, 10 and 105 families in dimensions 1, 2 and 3 respectively, while in higher dimensions the number of Fano families is yet unknown. An important problem is compactifying these families into moduli spaces via K-stability. In this talk, I will describe the compactification of the family of Fano threefolds, which is obtained by blowing up the projective space along a complete intersection of two quadrics which is an elliptic curve, into a K-moduli space using Geometric Invariant Theory (GIT). A more interesting setting occurs in the case of pairs of varieties and a hyperplane section where the K-moduli compactifications tessellate depending on a parameter. In this case it has been shown recently that the K-moduli decompose into a wall-chamber decomposition depending on a parameter, but wall-crossing phenomena are still difficult to describe explicitly. Using GIT, I will describe an explicit example of wall-crossing in the K-moduli spaces, where both variety and divisor differ in the deformation families before and after the wall, given by log pairs of Fano surfaces of degree 4 and a hyperplane section.