Perfect obstruction theory

Consider a regular embedding ${\displaystyle i:Y\to W}$ 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 ${\displaystyle [X,E^{\bullet }]=i^{!}[V]}$

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).

The previous construction works too with Deligne–Mumford stacks.