# Virtual Fundamental Classes

We have constructed obstruction theories of some stacks in Chapter 5. Under some assumption, they are perfect and give virtual fundamental classes. We will study their property in this chapter.

In Section 6.1, we obtain virtual fundamental classes for some stacks, by showing the perfectness of the obstruction theories. We compare the virtual fundamental classes of moduli stacks of δ-stable oriented reduced L-Bradlow pairs and δ-stable *L*-Bradlow pairs in Section 6.2. Although the moduli stacks are isomorphic up to étale proper morphisms, the obstruction theories are not the same in general, and we obtain the vanishing of the virtual fundamental class of the moduli of δ-stable *L*-Bradlow pairs in the case p_{g} = dim *H* ^{2}(*X,O* _{ X })> 0.

In Section 6.3, we study the virtual fundamental classes of moduli stacks of objects with rank one. In Subsection 6.3.1, we look at a moduli of *L*-abelian pairs. In particular, we give a detailed description of the virtual fundamental class when \(H^2(X,O) \neq 0 {\rm and} H^1(X,O) = 0\) are satisfied. In Subsection 6.3.3, we study the obstruction theory of parabolic Hilbert schemes. In the rest of this section, we show the splitting stated in Proposition 6.3.8.

In Sections 6.4–6.6, we give some relations of the virtual fundamental classes of some moduli stacks.

## Keywords

Line Bundle Commutative Diagram Cohomology Class Free Resolution Natural Morphism## Preview

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