Perfect obstruction theory

In algebraic geometry, given a Deligne–Mumford stack "X", a perfect obstruction theory for "X" consists of:
The notion was introduced by for an application to the intersection theory on moduli stacks; in particular, to define a virtual fundamental class.
Examples.
Schemes.
Consider a regular embedding formula_7 fitting into a cartesian square
where formula_9 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 formula_13 coming from the maps formula_14 and formula_15. Note that the associated virtual fundamental class is formula_16
Example 1.
Consider a smooth projective variety formula_17. If we set formula_18, then the perfect obstruction theory in formula_19 is
and the associated virtual fundamental class is
In particular, if formula_22 is a smooth local complete intersection then the perfect obstruction theory is the cotangent complex (which is the same as the truncated cotangent complex).
Deligne–Mumford stacks.
The previous construction works too with Deligne–Mumford stacks.
Symmetric obstruction theory.
By definition, a symmetric obstruction theory is a perfect obstruction theory together with nondegenerate symmetric bilinear form.
Example: Let "f" be a regular function on a smooth variety (or stack). Then the set of critical points of "f" carries a symmetric obstruction theory in a canonical way.
Example: Let "M" be a complex symplectic manifold. Then the (scheme-theoretic) intersection of Lagrangian submanifolds of "M" carries a canonical symmetric obstruction theory.