  This paper gives a canonical construction, in terms of additive cohomological
functors, of the universal formal deformation of a compact complex manifold
without vector fields (more generally of a faithful $g$-module, where $g$ is a
sheaf of Lie algebras without sections). The construction is based on a certain
(multivariate) Jacobi complex $J(g)$ associatd to $g$: indeed ${\mathbb
C}\oplus {\mathbb H}^0(J(g))^*$ is precisely the base ring of the universal
deformation.
