  We consider the compactification of the dual form of $N=1$ $D=10$
supergravity on a six-dimensional Calabi-Yau manifold. An $N=1$ off-shell
supergravity effective Lagrangian in four dimensions can be constructed in a
dual version of the gravitational sector (new-minimal supergravity form).
Superspace duality has a simple interpretation in terms of Poincar\'{e} duality
of two-form cohomology. The resulting $4D$ Lagrangian may describe the
low-energy point-field limit of a five-brane theory, dual to string theory,
provided Calabi-Yau spaces are consistent vacua of such dual theory.
