On complex manifolds, we can consider various holomorphic geometric structures such as affine structures, projective structures and conformal structures Introducing new characteristic forms defined by projective Weyl curvature tensors, we shall prove fomulae which relate them with Chern forms. As byproducts, we obtain relations among Chern es on complex manifolds with such geometric structures. Some of them give refinements of results by R. Gunning, S. Kobayashi and T. Ochiai.