Stratifications and quasi-projective coarse moduli spaces for the stack of Higgs bundles

Abstract

The classification problem for Higgs bundles of a fixed rank and degree on a compact Riemann surface is encoded in the moduli stack of such objects. Nitsure's GIT construction of the moduli space of semistable Higgs bundles produces a quasi-projective coarse moduli space (if the rank and degree are coprime) for the semistable stratum of this moduli stack. Nevertheless, GIT cannot be used to produce coarse moduli spaces for the complement of the semistable stratum. In this paper we use a recent generalisation of GIT, called Non-Reductive GIT, to construct two stratifications of the stack of Higgs bundles which satisfy the property that each stratum admits a quasi-projective coarse moduli space with an explicit projective completion. The first is a refinement of the Higgs Harder-Narasimhan stratification of the stack of Higgs bundles (defined by the instability type of the Higgs bundle), while the second is a refinement of the Harder-Narasimhan stratification (defined by the instability type of the underlying bundle). We provide a complete and explicit moduli-theoretic description of both refined stratifications in the rank 2 case.

Smoothness of non-reductive fixed point sets and cohomology of non-reductive GIT quotients

“…This is the case for moduli spaces of unstable vector or Higgs bundles on a smooth projective curve (and more generally of sheaves or Higgs sheaves on a smooth projective variety), which can be constructed using Non-Reductive GIT. In the Non-Reductive GIT set-up for the construction of these moduli spaces, Z min //R λ is a moduli space for those unstable bundles which are isomorphic to their Harder-Narasimhan graded (see [32,27]). In Remark 5.9 below we discuss the application of Theorem 5.7 to this example.…”

“…Examples of such moduli spaces are moduli spaces for unstable vector or Higgs bundles on a smooth projective curve, and more generally (Higgs) sheaves on a smooth projective variety, with a Harder-Narasimhan type of length two (this includes the case of rank two unstable vector or Higgs bundles). Indeed such moduli spaces can be constructed using Non-Reductive GIT (see [32,27]), a construction which requires performing the non-reductive blow-ups as the condition (ss = s = ∅[ U ]) is not satisfied.…”

“…Theorem B can be used to calculate the Poincaré series of moduli spaces which can be constructed as non-reductive GIT quotients of a smooth complex projective variety X by the action of a group H of the form given in Theorem A. This is the case for moduli spaces of unstable vector or Higgs bundles on a smooth projective curve (and more generally of unstable sheaves or Higgs sheaves on a smooth projective variety) with a fixed coprime Harder-Narasimhan type of length two 8 , as constructed in [12,32,27]. In this setting, all of the conditions of Theorem A are satisfied and thus the Poincaré series of these moduli spaces can be computed from the Poincaré series of Z min //R λ and of iterated blow-ups of it using (iii) of Theorem B 9 .…”

Quotients by Parabolic Groups and Moduli Spaces of Unstable Objects

“…Our primary motivating examples come from moduli of objects in a linear abelian category, where moduli spaces of semistable objects are constructed by reductive GIT and the HKKN stratification is closely related to a Harder-Narasimhan (or Shatz [34]) stratification; for example, this is the case for moduli of bundles or sheaves [23,22], Higgs sheaves [17] and quiver representations [21]. In these cases, constructing quotients of the unstable strata S β would give rise to moduli spaces of objects of fixed HN type; this has been described for certain length 2 HN filtrations in [10,17,25]. 1.1. Moduli of Unstable Objects.…”

“…Hence, [L] 0 fails for HN filtrations of length l > 2; this happens for moduli of (Higgs) sheaves and quiver representations. For HN filtrations of length l = 2, the problem of constructing moduli spaces of fixed HN type for sheaves and Higgs bundles are considered in [25] and [17]. 1.2. Quotienting-in-Stages.…”