Cohomology and Obstructions I: Geometry of formal Kuranishi theory

The principle "ambient cohomology of a Kaehler manifold annihilates obstructions" has been known and exploited since pioneering work of Kodaira. This paper extends and unifies many known results in two contexts, abstract deformations of compact Kaehler manifolds and deformations of submanifolds within a given deformation of the ambient manifold.