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One of my favorite explanation of the cohomology groups of low degree is that they arise as the automorphism group, tangent space and obstruction space (or where the obstruction lives) of a certain deformation problem.

My question is, is it possible to reverse the process of going from deformation problem to cohomology groups? Say I have H^0, H^1 and H^2 of a certain sheaf, is it possible to find a deformation problem such that these groups control, i.e. such that they arises as the Automorphism-Tangent-Obstruction of that deformation problem?

To be more concrete, say we have a smooth scheme X over k, and our sheaf is the tangent sheaf T_X of X, how do I "find" the problem of "smooth deformation of X"?

(I know I have been pretty vague. I know how to interpret cohomology classes as gerbes/torsors, but I'm not quite satisfied (I like these interpretations though). I want a way to find deformation problem that produces gerbes as obstructions...)

Quote from wikipedia: Reverse engineering is the process of discovering the technological principles of a human made device, object or system through analysis of its structure, function and operation. [I hope my usage of this phrase is correct.]

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A related question is… --- you need more structure than the cohomology groups, usually a differential graded Lie algebra structure lifting them. This will allow you to reconstruct the (germ of the) moduli space, but not the universal family -- ie you won't know WHAT you're deforming, just how to parametrize the deformations.. –  David Ben-Zvi Apr 11 '11 at 3:16

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In this generality, I would say that this is not possible: the same cohomology can be responsible for controlling quite different deformations problems.

Just an example: in formal deformation quantization you have the dgla of multivector fields "controlling" both, the classical deformation problem of deforming a Poisson structure, and second, by the (highly non-trivial) formality theorem, the deformation problem of quantising the algebra of smooth functions. Even the moduli spaces of the deformation problems are the same...

So two deformation problem of quite different nature arise from the Schouten algebra.

So little cousing of this is perhaps easier to understand: on a symplectic manifold the (abelian) dgla of differential forms controlls the deformation theory of the symplectic form up to formal diffeos, the (formal series in the) second deRham cohomology yields the parametrization of the moduli space. But also the star products quantizing the symplectic form are controlled by this: the equivalence classes of formal star products are again formal series in the second deRham cohomology. Of course, this is just a special case of the above Kontsevich classification, but there are much simpler proofs in this case.

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Good point. More extremely, take the dgla which is just an n-dimensional vector space in degree one and the rest zero. It controls the deformation theory of any variety, or algebra, or elephant, with n-dimensional deformations, no infinitesimal automorphisms and no obstructions.. –  David Ben-Zvi Apr 11 '11 at 14:37

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