Let \(X\) be a complex projective scheme and \(G\) a complex reductive group acting linearly on \(X\) with respect to an ample line bundle. According to Mumford’s Geometric Invariant Theory (GIT), there is a projective scheme \(X/\!/G\) which is a categorical quotient of the open subscheme \(X^{\mathrm{ss}}\) of \(X\) consisting of points which are semistable for the given linearised action. The points of \(X/\!/G\) correspond to S-equivalence classes of semistable points. In [Mathematical Notes, 31, Princeton, New Jersey: Princeton University Press (1984); Zbl 0553.14020], the second author defined a stratification \(\{S_\beta:\beta\in{\mathcal B}\}\) of \(S\) into disjoint \(G\)-invariant locally closed subschemes, one of which is \(X^{\mathrm{ss}}\). This stratification depends on the lineaarisation and a choice of invariant inner product on the Lie algebra of a maximal compact subgroup of \(G\). It can be defined also using the results of G. Kempf and L. Ness [Lect. Notes in Math. 732, 233–243 (1979; Zbl 0407.22012)].
In this paper, the authors consider the problem of constructing a quotient for each unstable stratum \(S_\beta\). One can define a linearisation for the action of \(G\) on a projective completion \(\hat{S}_\beta\) of \(S_\beta\) and this canonical linearisation provides a categorical quotient for \(S_\beta\). However, the linearisation is not ample and the quotient collapses more orbits than one would like. To resolve this problem, the authors consider small perturbations of the canonical linearisation.
This idea is then applied to the construction of moduli spaces of sheaves of fixed Harder-Narasimhan type with some extra data. Suppose that \({\mathcal F}\) has Harder-Narasimhan filtration \(0\subset{\mathcal F}^{(1)}\subset\cdots\subset{\mathcal F}^{(s)}={\mathcal F}\) and let \({\mathcal F}_i:={\mathcal F}^{(i)}/{\mathcal F}^{(i-1)}\). The Harder-Narasimhan type of \({\mathcal F}\) is the \(n\)-tuple \((P_1,\dots,P_s)\), where \(P_i\) is the Hilbert polynomial of \({\mathcal F}_i\). A sheaf \({\mathcal F}\) is said to be \(\tau\)-compatible if it possesses a filtration as above, where now the possibility that \({\mathcal F}^{(i)}={\mathcal F}^{(i-1)}\) is allowed. An \(n\)-rigidification for \({\mathcal F}\) is then an isomorphism \(H^0({\mathcal F}(n))\cong\oplus H^0({\mathcal F}_i(n))\) with the obvious compatibility relations. A concept of \(\theta\)-semistability for suitable \(\theta\in{\mathbb Q}^s\) can be defined and the following theorem (Theorem 8.9) proved. Let \(W\) be a projective scheme over \(\mathbb{C}\) and \(\tau\) a fixed Harder-Narasimhan type. For \(\theta\in{\mathbb Q}^s\) and \(n\gg0\), there is a projective scheme \(M^{\theta-ss}(W,\tau,n)\) which corepresents the moduli functor of \(\theta\)-semistable \(n\)-rigidified sheaves of Harder-Narasimhan type \(\tau\) over \(W\). The points of \(M^{\theta-ss}(W,\tau,n)\) correspond to S-equivalence classes of \(\theta\)-semistable \(n\)-rigidified sheaves with Harder-Narasimhan type \(\tau\).
The construction of the stratification \(\{S_\beta\}\) for \(X\) a smooth projective variety is recalled in section 2, followed by the construction of quotients in section 3 and the extension to an arbitrary projective scheme in section 4. Simpson’s construction of the moduli of semistable sheaves is recalled in section 5. In section 6, a stratification by Harder-Narasimhan types is described. The concept of an \(n\)-rigidified sheaf is introduced in section 7, followed by the construction of the moduli spaces in section 8.