Quotients of unstable subvarieties and moduli spaces of sheaves of fixed Harder-Narasimhan type

Abstract

When a reductive group G acts linearly on a complex projective scheme X, there is a stratification of X into G-invariant locally closed subschemes, with an open stratum X ss formed by the semistable points in the sense of Mumford's geometric invariant theory which has a categorical quotient X ss → X//G. In this article, we describe a method for constructing quotients of the unstable strata. As an application, we construct moduli spaces of sheaves of fixed Harder-Narasimhan type with some extra data (an 'n-rigidification') on a projective base.

Stratifications for moduli of sheaves and moduli of quiver representations

“…Theorem 5.19 (see also [11]). Let τ be a HN type; then, for m >> n >> 0, the Shatz stratum Q n,τ is a closed subscheme of the Hesselink stratum S βn,m(τ ) in Quot(V n ⊗ O X (−n), P ).…”

“…In this section, for fixed n, we describe this stratification of Quot := Quot(V n ⊗ O X (−n), P ); this completes the partial description of the Hesselink stratification on R n given in [11].…”

“…In this article, we complete the description of the Hesselink stratification of Quot n and extend the inclusion result of [11] to non-pure HN types (cf. Theorem 5.19).…”

Graded linearisations

“…In particular we can consider moduli spaces of sheaves of fixed Harder-Narasimhan type over a nonsingular projective variety W (cf. [23]). There are well known constructions going back to Simpson [39] of the moduli spaces of semistable pure sheaves on W of fixed Hilbert polynomial as GIT quotients of linear actions of suitable special linear groups G on schemes Q (closely related to quot-schemes) which are G-equivariantly embedded in projective spaces.…”

Stratifying Quotient Stacks and Moduli Stacks

“…where [V n /G n ] is the quotient stack associated to a linear action on a quasi-projective scheme V n by a group G n which is reductive (or more generally has internally graded unipotent radical). We can look for suitable 'stability conditions' on M: linearisations (L n ) n 0 for the actions of G n on projective completions V n of V n and invariant inner products on Lie G n which are compatible in the sense that the stratification induced by L n on [V n /G n ] restricts to the stratification induced by L m on [V m /G m ] when n > m. This situation arises for sheaves over a projective scheme [20,6], for example, and also for projective curves [21], and we obtain a stratification {Σ γ |γ ∈ Γ} of the stack M such that each stratum Σ γ is isomorphic to a quotient stack [W γ /H γ ], where W γ is quasi-projective acted on by a linear algebraic group H γ with internally graded unipotent radical, and there is a geometric quotient W γ /H γ which is a coarse moduli space for Σ γ . The geometric quotient W γ /H γ will be projective if semistability coincides with stability in an appropriate sense for the action of H γ on a suitable projective completion of W γ with respect to an induced linearisation.…”