Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves (Q1321792)

Let \(f:X \to S\) be a smooth, projective, geometrically integral morphism of locally noetherian schemes, \(D\) be an effective relative Cartier divisor on \(X/S\) and let \({\mathcal O}_ X(1)\) be an \(f\)-very ample invertible sheaf. A parabolic sheaf on a geometric fibre \(X_ S\) of \(f\) is a triple \((E,F_ *, \alpha_ *)\) consisting of a torsion free coherent sheaf \(E\), a filtration \[ E = F_ 1(E) \supset F_ 2(E) \supset \cdots \supset F_ l(E) \supset F_{l+1} = E(-D) \] and a system of weights \(0 \leq \alpha_ 1 < \alpha_ 2 < \cdots < \alpha_ l<1\). The author constructs a moduli scheme of equivalence classes of parabolic semi-stable sheaves and shows that it is projective over \(S\) under some boundedness conditions. In fact the method leads to a construction of a moduli scheme of ``parabolic pairs''. Namely, let \(\Omega\) be a locally free \({\mathcal O}_ X\)-module. A pair \((E_ X, \varphi)\) of a parabolic sheaf \(E_ X\) is said to be a parabolic \(\Omega\)-pair if \(\varphi \wedge \varphi = 0\) where \(\varphi \wedge \varphi\) is the following homomorphism \(E@>\varphi>>E \otimes_ X \Omega@>\varphi \otimes 1>>E \otimes_ X \Omega\otimes_ X \Omega \to E \otimes_ X \bigwedge^ 2 \Omega\) and \(E \otimes_ X \Omega\) is a parabolic sheaf such that \((E \otimes_ X \Omega)_ \alpha = E_ \alpha \otimes_ X \Omega\). Combining the notion of parabolic sheaves and that of \(\Omega\)-pairs one comes to a notion of parabolic \(\Omega\)-pairs. The word ``parabolic Higgs sheaves'' means \(\Omega^ 1_ X (\log D)\)-pairs. The main theorem states the existence of a moduli scheme of equivalence classes of parabolic semi- stable \(\Omega\)-pairs.