String gravitational equations with Hermitian structure

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.