Abstract

Geometric invariant theory (GIT) is the quotient theory of algebraic geometry, which applies to group actions on varieties together with an ample line bundle. The main point of GIT is that only the “stable” orbits are parametrised in the quotient.

In applications, the ampleness hypothesis frequently fails, or is at least not known to hold. I will describe a generalisation of some of the ideas of GIT to more general line bundles, where one writes the given line bundle as the difference of a pair of ample line bundles. The most interesting aspect is that the traditional numerical characterisation of stability is known to be false in this general setting, so to give a numerical characterisation, some new geometric input is needed.

I will then describe some applications to Kähler geometry, such as a new proof that a version of K-stability of a smooth Fano variety is equivalent to the existence of a Kähler-Einstein metric.

This is all joint work with Rémi Reboulet.