GIT-characterization of Harder-Narasimhan filtrations

Abstract. We will discuss constructions of moduli spaces in algebraic geometry by using Geometric Invariant Theory (GIT). 

When performing such constructions we usually impose a notion of stability for the objects we want to classify and another notion of GIT stability appears, then it is shown that both notions coincide. For an object which is unstable (i.e. contradicting the stability condition) there exists a unique canonical filtration, called the Harder-Narasimhan filtration. On the other hand, GIT stability is checked by 1-parameter subgroups by the classical Hilbert-Mumford criterion, and it turns out that there exists a unique 1-parameter subgroup giving a notion of maximal unstability in the GIT sense. 

We show how to prove that this special 1-parameter subgroup can be converted into a filtration of the object and coincides with the Harder-Narasimhan filtration, hence both notions of maximal unstability are the same. We will present the correspondence for the moduli problem of classifying coherent sheaves on a smooth complex projective variety. 

A similar treatment can be used to prove the analogous result for other moduli problems: holomorphic pairs, Higgs sheaves, rank 2 tensors and finite dimensional quiver representations.