Obstruction Theories of Moduli Stacks and Master Spaces

Let X be a smooth connected projective surface with a base point over an algebraically closed field k of characteristic 0. Let D be a smooth hypersurface of X. In this chapter, we study obstruction theories of moduli stacks of some kinds of stable objects on X with parabolic structure along D. The naive strategy for construction was explained in Subsection 2.4.2.We will also discuss obstruction theories for master spaces.

In Section 5.1, we study an obstruction theory associated to a torsion-free sheaf E on U × X, where X is a smooth projective surface. We put

$$\mathop{g}(V_\bullet):=Hom (V_\bullet, V_\bullet)^{\vee} [-1], \quad {\rm Ob}(V_\bullet):=RpX_{\ast}\left(\mathop{g}(V-\bullet)\otimes \omega_{X}\right)$$

for a resolution $V_\bullet$ of E. In Subsection 5.1.1, we explain how to obtain the morphisms $g(V_\bullet)\rightarrow L_{U\times X/X}$ and ${\rm Ob}(V_\bullet)\rightarrow L_U$. In Subsection 5.1.2, we observe that $\mathop{g}(V_\bullet)$ is decomposed into the trace-free part and the diagonal part, and that the diagonal part is related to the determinant bundle. In Subsection 5.1.3, we give some factorization which will be useful for construction of obstruction theories of master spaces. In Subsection 5.1.4, we give an obstruction theory of the open subset of a moduli stack of torsion-free sheaves determined by the condition O_m, by directly applying the construction in Subsection 5.1.1. As a special case, we look at an obstruction theory of the moduli of line bundles in Subsection 5.1.5, which will be used in the construction of a relative obstruction theory for orientations in Section 5.2.

In Section 5.3, we study a relative obstruction theory for L-sections. In Subsections 5.3.1–5.3.2, we construct a complex with a morphism, and show its relative obstruction property. See also Subsection 5.3.5 for another construction, which might be useful for simplification. In Subsection 5.3.3, we give a factorization which will be useful to construct obstruction theories of master spaces. In Section 5.4, we argue a relative obstruction theory for reduced L-sections. We need some modification to the construction in Section 5.3. In Section 5.5, we study a relative obstruction theory for parabolic structures. By pulling them together, we can construct obstruction theories of moduli stacks considered in this monograph, which is explained in Section 5.6.

Then, we construct obstruction theories of master spaces in Section 5.7. Once we have factorizations as in Subsections 5.1.3, 5.3.3 and 5.4.3, the construction is easy. We also obtain obstruction theories for some related stacks. In Subsections 5.7.6–5.7.7, we give only the statements in some easier cases for explanation.

In Section 5.8, we investigate obstruction theories of the fixed point sets. In Section 5.9, we study equivariant obstruction theories of master spaces and the induced obstruction theory of the fixed point sets.