In algebraic geometry, given a Deligne–Mumford stack ''X'', a perfect obstruction theory for ''X'' consists of: # a perfect two-term complex

E =^ \to E^0 /math> in the derived category D(\text(X)_) of quasi-coherent étale sheaves on ''X'', and
# a morphism \varphi\colon E \to \textbf_X, where \textbf_X is the cotangent complex of ''X'', that induces an isomorphism on h^0 and an epimorphism on h^.

The notion was introduced by  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
X & \xrightarrow & V \\
g \downarrow & & \downarrow f \\
Y & \xrightarrow & W
\end

where

V,W

are smooth. Then, the complex :

E^\bullet =^*N_^ \to j^*\Omega_V /math> (in degrees -1, 0)
forms a perfect obstruction theory for ''X''. The map comes from the composition
: g^*N_^\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_X^\bullet coming from the maps g^*\mathbf_Y^\bullet \to \mathbf_X^\bullet and j^*\mathbf_V^\bullet \to \mathbf_X^\bullet . Note that the associated virtual fundamental class is,E^\bullet = i^! /math>

Example 1

Consider a smooth projective variety

Y \subset \mathbb^n

. If we set

V = W

, then the perfect obstruction theory in

D^(X)

is :

_^\vee \to \Omega_/math>
and the associated virtual fundamental class is
:,E^\bullet = i^! mathbb^n /math>
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.

Notes

References

* * * {{Cite web, url=https://mathoverflow.net/q/211932 , title=Understanding the obstruction cone of a symmetric obstruction theory, last=Oesinghaus, first=Jakob , website= MathOverflow, date=2015-07-20, access-date=2017-07-19