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 in the derived category of quasi-coherent étale sheaves on X, and
  2. a morphism , where is the cotangent complex of X, that induces an isomorphism on and an epimorphism on .

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

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Schemes

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Consider a regular embedding   fitting into a cartesian square

 

where   are smooth. Then, the complex

  (in degrees  )

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

 

This is a perfect obstruction theory because the complex comes equipped with a map to   coming from the maps   and  . Note that the associated virtual fundamental class is  

Example 1

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Consider a smooth projective variety  . If we set  , then the perfect obstruction theory in   is

 

and the associated virtual fundamental class is

 

In particular, if   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

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The previous construction works too with Deligne–Mumford stacks.

Symmetric obstruction theory

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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

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References

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  • Behrend, Kai (2005). "Donaldson–Thomas invariants via microlocal geometry". arXiv:math/0507523v2.
  • Behrend, Kai; Fantechi, Barbara (1997-03-01). "The intrinsic normal cone". Inventiones Mathematicae. 128 (1): 45–88. arXiv:alg-geom/9601010. Bibcode:1997InMat.128...45B. doi:10.1007/s002220050136. ISSN 0020-9910. S2CID 18533009.
  • Oesinghaus, Jakob (2015-07-20). "Understanding the obstruction cone of a symmetric obstruction theory". MathOverflow. Retrieved 2017-07-19.

See also

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