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 → E 0 ] {\displaystyle E=[E^{-1}\to E^{0}]} in the derived category D ( Qcoh ( X ) e t ) {\displaystyle D({\text{Qcoh}}(X)_{et})} of quasi-coherent étale sheaves on X, and a morphism φ : E → L X {\displaystyle \varphi \colon E\to {\textbf {L}}_{X}} , where L X {\displaystyle {\textbf {L}}_{X}} is the cotangent complex of X, that induces an isomorphism on h 0 {\displaystyle h^{0}} and an epimorphism on h − 1 {\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 I : Y → W {\displaystyle I\colon Y\to W} fitting into a cartesian square
X → j V g ↓ ↓ f Y → i W {\displaystyle {\begin{matrix}X&{\xrightarrow {j}}&V\\g\downarrow &&\downarrow f\\Y&{\xrightarrow {i}}&W\end{matrix}}}
where V , W {\displaystyle V,W} are smooth. Then, the complex
E ∙ = [ g ∗ N Y / W ∨ → j ∗ Ω V ] {\displaystyle E^{\bullet }=[g^{*}N_{Y/W}^{\vee }\to j^{*}\Omega _{V}]} (in degrees − 1 , 0 {\displaystyle -1,0} ) forms a perfect obstruction theory for X. The map comes from the composition
g ∗ N Y / W ∨ → g ∗ i ∗ Ω W = j ∗ f ∗ Ω W → j ∗ Ω V {\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 L X ∙ {\displaystyle \mathbf {L} _{X}^{\bullet }} coming from the maps g ∗ L Y ∙ → L X ∙ {\displaystyle g^{*}\mathbf {L} _{Y}^{\bullet }\to \mathbf {L} _{X}^{\bullet }} and j ∗ L V ∙ → L X ∙ {\displaystyle j^{*}\mathbf {L} _{V}^{\bullet }\to \mathbf {L} _{X}^{\bullet }} . Note that the associated virtual fundamental class is [ X , E ∙ ] = i ! [ V ] {\displaystyle [X,E^{\bullet }]=i^{!}[V]}
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