The early ideas appeared already in the work of Kodaira and Spencer on the deformation of complex structures. The equations describing a complex structure are non-linear over-determined PDEs. One way to study the deformations of a given complex structure is to linearize these equations and then try to complete a linear solution to a 1-parameter family of non-linear (true) deformations. As Kodaira and Spencer discovered, not all linear solutions can be continued to non-linear deformations. A problem can appear as follows. Solving the defining equations of a complex structure perturbatively, at the quadratic order one needs to solve a linear inhomogeneous equation with a source quadratic in the linear order solution. If this source lies outside the range of the linear operator in the quadratic order equation, there is obviously no solution and the linear order solution cannot be continued to a non-linear one (not even to second order). In other segments of the PDE literature, this phenomenon is called linearization instabilitylinearization instability. So, some obstructions to continuing the linear order deformation of a complex structure to a $1$-parameter family of non-linear deformations lies in the discrepancy between the range of the linear differential operator appearing at the quadratic perturbative order and the non-linear map that is used to construct its source term from the linear order solution. Kodaira and Spencer figured out that this discrepancy is absent if some cohomology of the compatibility complex of the linearized defining equations of a complex structure vanishes. The same cohomology could also be described as the cohomology of the sheaf of complex vector fields on the underlying complex manifold. All of this is described in Kodaira's book:
