A Non-supersymmetry Conditions On The Tight Three-dimensional Complex Manifold

We first introduce a method proposed by Calabi to construct non-Kahler complex three dimensional manifolds. We aslo recall the supersymmetry equa-tion in superstring theory. Following Calabi's method, one can construct a com-plex torus bundle on Catenoid. On such a manifold, we construct explicitly a non-vanishing holomorphic three form and then use this form to construct a family of conformal balanced metrics which satisfy the supersymmetry condition.