Harder–Narasimhan filtration for rank 2 tensors and stable coverings

Abstract

We construct a Harder-Narasimhan filtration for rank 2 tensors, where there does not exist any such notion a priori, as coming from a GIT notion of maximal unstability. The filtration associated to the 1-parameter subgroup of Kempf giving the maximal way to destabilize, in the GIT sense, a point in the parameter space of the construction of the moduli space of rank 2 tensors over a smooth projective complex variety, does not depend on a certain integer used in the construction of the moduli space, for large values of the integer. Hence, this filtration is unique and we define the Harder-Narasimhan filtration for rank 2 tensors as this unique filtration coming from GIT. Symmetric rank 2 tensors over smooth projective complex curves define curve coverings lying on a ruled surface, hence we can translate the stability condition to define stable coverings and characterize the Harder-Narasimhan filtration in terms of intersection theory.

Geometric Invariant Theory, holomorphic vector bundles and the Harder-Narasimhan filtration

“…We consider the git, hvb & hnf construction of a moduli space for these objects by King [Ki] and associate to an unstable representation an unstable point, in the sense of GIT, in a parameter space where a group acts. Then, the 1-parameter subgroup gives a filtration of subrepresentations and we prove that it coincides with the Harder-Narasimhan filtration for that representation (see [Za3] for quiver representations and [Za1, Chapter 3] for Q-sheaves).…”

“…where E is a rank 2 coherent torsion free sheaf over a smooth complex projective variety X and M a line bundle over X. In this case, the Harder-Narasimhan filtrations are simply line subbundles L. In [Za3], symmetric rank 2 tensors are interpreted as degree s coverings X → X lying on the ruled surface P(E), to define a notion of stable covering and characterize geometrically the maximally destabilizing subbundle L ⊂ E in terms of intersection theory and configurations of points as in Example 3.13. There are also other categories, such as quiver representations, where these ideas apply.…”

“…GIT unstable points are localized as those contradicting the criterion, analogous to unstable bundles being the objects not verifying the stability condition for bundles. The result of Kempf [Ke] and the well known Harder-Narasimhan filtration [HN] show the two sides of this maximal principle which are put in correspondence in [Za3,GSZ1], and which is linked to symplectic and differencial geometry in [GRS]. For a modern treatment generalizing the structure of unstability to quotient stacks problems see Halpern-Leistner theory [H-L].…”

On the Gieseker Harder–Narasimhan filtration for principal bundles

“…These ideas give a method to construct the Harder-Narasimhan filtration in cases where we do not know it a priori. The paper [Za1] applies this to rank 2 tensors and the method does not work in more general situations (c.f. [Za2, Section 2.5]).…”

A GIT interpretation of the Harder–Narasimhan filtration

“…In other words, our method gives a different proof of the existence of the Harder-Narasimhan filtration, and in principle our method could be used to define the Harder-Narasimhan filtration (using the Kempf filtration for m large) in a moduli problem where there is still no Harder-Narasimhan filtration known. This is the case of [Za1], where a similar construction is developed for rank 2 tensors.…”