Perfect obstruction theory
# Perfect obstruction theory

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

* a perfect two-term complex $E=[E^{-1}\to E^{0}]$ in the derived category $D({\text{Qcoh}}(X)_{et})$ of quasi-coherent étale sheaves on X, and
* a morphism $\varphi \colon E\to {\textbf {L}}_{X}$, where ${\textbf {L}}_{X}$ is the cotangent complex of X, that induces an isomorphism on $h^{0}$ and an epimorphism on $h^{-1}$.

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.

## Examples

### Schemes

Consider a regular embedding $I\colon Y\to W$ fitting into a cartesian square

    ${\begin{matrix}X&{\xrightarrow {j}}&V\\g\downarrow &&\downarrow f\\Y&{\xrightarrow {i}}&W\end{matrix}}$

where $V,W$ are smooth. Then, the complex

    $E^{\bullet }=[g^{*}N_{Y/W}^{\vee }\to j^{*}\Omega _{V}]$ (in degrees $-1,0$)

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

    $g^{*}N_{Y/W}^{\vee }\to g^{*}i^{*}\Omega _{W}=j^{*}f^{*}\Omega _{W}\to j^{*}\Omega _{V}$

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

**Example 1**

Consider a smooth projective variety $Y\subset \mathbb {P} ^{n}$. If we set $V=W$, then the perfect obstruction theory in $D^{[-1,0]}(X)$ is

    $[N_{X/\mathbb {P} ^{n}}^{\vee }\to \Omega _{\mathbb {P} ^{n}}]$

and the associated virtual fundamental class is

    $[X,E^{\bullet }]=i^{!}[\mathbb {P} ^{n}]$

In particular, if $Y$ 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.

