  We consider a string model at one-loop related to a $\sigma$-model whose
antisymmetric tensor field is constructed as complex structure on the
background manifold, specially on a manifold $R\times N$ where $N$ is a complex
manifold. As an example, we consider a homogeneous anisotropic
$(1+4)$-dimensional $\sigma$-model where space part of the background is a
$4$-dimensional complex manifold. By solving the related one-loop
$\beta$-functions we obtain a static solution so that by reduction of this
solution to $(1+3)$-dimension we obtain a static solution of Einstein equation
where the matter sector is effectively interpreted as an inhomogeneous,
anisotropic and barotropic matter satisfying all the energy conditions.
Finally, the $T$-dual background of the solution is investigated and it is
shown that the duality transformation and reduction processes commute with each
other.
