In algebraic geometry, given a Deligne–Mumford stack X, a perfect obstruction theory for X consists of: The notion was introduced by Kai Behrend and Barbara Fantechi (1997) for an application to the intersection theory on moduli stacks; in particular, to define a virtual fundamental class.
Then, the complex forms a perfect obstruction theory for X.
[1] The map comes from the composition This is a perfect obstruction theory because the complex comes equipped with a map to
Note that the associated virtual fundamental class is
Consider a smooth projective variety
is and the associated virtual fundamental class is In particular, if
is a smooth local complete intersection then the perfect obstruction theory is the cotangent complex (which is the same as the truncated cotangent complex).
The previous construction works too with Deligne–Mumford stacks.
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.
Then the (scheme-theoretic) intersection of Lagrangian submanifolds of M carries a canonical symmetric obstruction theory.