Perfect obstruction theory

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

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

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 ${\displaystyle I\colon Y\to W}$ fitting into a cartesian square

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

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

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

forms a perfect obstruction theory for X.^[1] The map comes from the composition

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

This is a perfect obstruction theory because the complex comes equipped with a map to ${\displaystyle \mathbf {L} _{X}^{\bullet }}$ 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 ${\displaystyle [X,E^{\bullet }]=i^{!}[V]}$

Example 1

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

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

and the associated virtual fundamental class is

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

In particular, if ${\displaystyle Y}$ 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.