# Preliminaries

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In Section 2.1, we prepare some convention. In Section 2.2, we review basic results from the geometric invariant theory. In particular, we recall a sufficient condition for a quotient stack to be Deligne-Mumford and proper.We also recall Mumford-Hilbert criterion, and look at some easy examples. The results will be used in Chapter 4.

In Section 2.3, we review some basic facts on cotangent complexes. Then, we recall how to express cotangent complexes of quotient stacks in Subsection 2.3.2, which will be used in Chapter 5 frequently. We also study some more examples in Subsection 2.3.3, which will be used in Sections 6.3, 6.4 and 6.6.

In Section 2.4, we review obstruction theory in the sense of K. Behrend-B. Fantechi [6]. We explain a naive strategy to construct obstruction theories of moduli stacks in Subsection 2.4.2. We recall an obstruction theory of locally free subsheaves in Subsection 2.4.3. It gives obstruction theories of moduli spaces of torsion-free quotient sheaves over a smooth projective surface. The result will be used in Section 5.6. We also obtain the smoothness of moduli spaces of quotient torsion-sheaves over a smooth projective curve, although we will not use it later. In Subsection 2.4.4, we recall an obstruction theory of filtrations of a vector bundle on a smooth projective curve. It will be used to construct a relative obstruction theory for quasi-parabolic structures.

In Section 2.5, we recall some standard results for equivariant complexes on Deligne-Mumford stacks with GIT construction, which will be used in Section 5.9. In Section 2.6, we give some elementary remarks on extremal sets, which are used in Sections 4.3–4.4. In Section 2.7, we give remarks on the twist of line bundles.

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