obstruction theories in algebraic geometry

Question

I'd like to know about the history of obstruction theories in algebraic geometry, as well as the relationship with concepts of the same name in topology. I would also like to know where obstruction theories have been used in algebraic geometry. The applications I know of are in Artin's criteria for showing that stacks are algebraic, and in the construction of virtual cycle classes in Gromov-Witten, Donaldson-Thomas, and similar theories.

Morally, an obstruction theory is supposed to control infinitesimal liftings: suppose $X \rightarrow Y$ is a map of algebraic geometry objects (schemes or stacks or whatever) and $S \subset S'$ is a square-zero extension of $Y$-schemes. An obstruction theory for $X$ over $Y$ is, vaguely speaking, a way of associating to any $Y$-morphism $S \rightarrow X$ an obstruction to the existence of an extension of that map to a $Y$-morphism $S' \rightarrow X$. In any individual lifting problem, an obstruction can usually be found in some cohomology group or other, but it is useful for some purposes (like the ones noted above) to have an abstract definition.

I know of four attempts to axiomatize the notion of an obstruction theory in algebraic geometry already:

1) Artin, M. Versal deformations and algebraic stacks

2) Fantechi, B. and Manetti, M. Obstruction calculus of functors of Artin rings, I

3) Li, J. and Tian, G. Virtual moduli cycles and Gromov--Witten invariants of algebraic varieties

4) Behrend, K. and Fantechi, B. The intrinsic normal cone

Answer 1

As far as I know, the prototypes of obstruction theories in algebraic geometry originated from the more general Kodaira-Spencer theory of deformation of complex manifolds [see Kodaira-Spencer, On deformations of complex analytic structures I-II-III, Annals of Math., 1958-1960].

The crucial question was

When an "infinitesimal" deformation of a compact complex manifold $M$ (in particular, of a complex projective variety) gives rise to a "genuine" deformation of $M$, i.e. a deformation over a disk?

The answer to this question is contained in the following result, see [Kodaira, Complex manifolds and deformations of complex structures, Theorem 5.1]:

Theorem. Suppose given a compact complex manifolds $M$, and $\theta \in H^1(M, \Theta_M)$. In order that there may exist a complex analytic family $\omega \colon \mathcal{M} \to B$ such that $\omega^{-1}(0)=M$ and $(dM_t/dt)_{t=0}=\theta$, it is necessary that $[\theta, \theta]=0$ holds.

In fact, Kodaira explicitly says that "if $[\theta, \theta] \neq 0$ there is no deformation $M_t$ with $\omega^{-1}(0)=M$ and $(dM_t/dt)_{t=0}=\theta$. In this sense, we call $[\theta, \theta] \in H^2(M, \Theta_M)$ the obstruction to the deformation of $M$".

Of course Kodaira was well aware that the condition $[\theta, \theta]=0$ is not sufficient in general, since there can be higher-order obstructions, corresponding to the need of finding higher and higher truncations of the solution of the Maurer-Cartan equation governing the deformation of the given complex structure. Only if all these obstructions vanish we can hope to find our complex analytic family $\mathcal{M}$.

In Kodaira's words, "Thus we have infinitely many obstructions to the deformations of $M$. In view of this fact we call $[\theta, \theta]$ the primary obstruction".

The obstruction theories coming later in algebraic geometry, as far as I know, were build up in order to rephrase and extend Kodaira-Spencer theory in a completely algebro-geometrical setting (for instance, making possible deformation theory in characteristic $p$), in order to deform objects different from complex structures, such as coherent sheaves, subvarieties, maps, and in order to understand the difference between the deformations in the analytic sense and those in the algebraic sense ("algebraization problem").

Answer 2

Fabien Morel has developed a version in $\mathbb{A}^1$-homotopy theory of the obstruction theory which is familiar in algebraic topology. See for example his paper http://www.mathematik.uni-muenchen.de/~morel/bgln.pdf . He uses it to prove some results about vector bundles on smooth affine varieties which are analogues of classical results about vector bundles on compact (real) manifolds.