Symmetric obstruction theory

In algebraic geometry, given a Deligne–Mumford stack X, a perfect obstruction theory for X consists of:

The notion was introduced by (Behrend–Fantechi 1997) for an application to the intersection theory on moduli stacks; in particular, to define a virtual fundamental class.

Consider a regular embedding ${\displaystyle i:Y\to W}$ fitting into a cartesian square

where ${\displaystyle V,W}$ are smooth. Then, the complex

forms a perfect obstruction theory for X. The map comes from the composition

This is a perfect obstruction theory because the complex comes equipped with a map to ${\displaystyle \mathbf {L} _{X}^{\bullet }}$ coming from the maps ${\displaystyle g^{*}\mathbf {L} _{Y}^{\bullet }\to \mathbf {L} _{X}^{\bullet }}$ and ${\displaystyle j^{*}\mathbf {L} _{V}^{\bullet }\to \mathbf {L} _{X}^{\bullet }}$. Note that the associated virtual fundamental class is ${\displaystyle [X,E^{\bullet }]=i^{!}[V]}$

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