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, especially on a manifold RxN 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 four-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.