Gergely Bérczi explains:

»Geometric Invariant Theory (GIT) was developed by Mumford in the 1960’s to provide a systematic description of quotients of algebraic varieties by reductive algebraic group actions. This fundamental theory quickly became an inescapable tool for mathematicians and physicists in classification problems, and it was recognised by Fields medal in 1974. The theory, however, fails at a fundamental level when the group acting is not reductive. As a result of a long-running program, with Frances Kirwan et al we extended Mumford’s GIT to a broad class of non-reductive groups.

In the paper ‘Moment maps and cohomology of non-reductive quotients’ with F. Kirwan, we define moment maps for non-reductive group actions and prove that our non-reductive GIT quotient is indeed canonical: it is diffeomorphic to the symplectic quotient corresponding to the moment map. This discovery led to a better understanding of the topology, and in particular the rational cohomology (Betti numbers, intersection theory) of these quotients.

Our non-reductive GIT theory provides a powerful new machinery to attack old, unsolved problems in complex geometry, singularity theory and enumerative geometry. 

In the paper ‘Non-reductive geometric invariant theory and hyperbolicity’ with Frances Kirwan we prove the Green-Griffiths-Lang (1985) and Kobayashi (1970) hyperbolicity conjectures for generic projective hypersurfaces of polynomial degree. These are landmark conjectures of complex geometry, significantly influencing the trajectory of the field over the past five decades. They embody the concept that varieties with sufficiently ample canonical bundle can not contain entire holomorphic curves.

In our work, following the strategy of Demailly, Siu, Diverio, Merker and Rousseau, we reduce the hyperbolicity problems to intersection theory of non-reductive quotients, where the results of 'Moment maps and cohomology of non-reductive quotients' provides an efficient computational toolbox.«

The papers are published in Inventiones Mathematicae via the following links:

• Moment maps and cohomology of non-reductive quotients
• Non-reductive geometric invariant theory and hyperbolicity