We investigate real local isometric immersions of Kähler manifolds $${\mathbb{C}Q^2_c}$$ of constant holomorphic curvature 4c into complex projective 3-space. Our main result is that the standard embedding of $${\mathbb{C}P^2}$$ into $${\mathbb{C}P^3}$$ has strong rigidity under the class of local isometric transformations. We also prove that there are no local isometric immersions of $${\mathbb{C}Q^2_c}$$ into $${\mathbb{C}P^3}$$ when they have different holomorphic curvature. An important method used is a study of the relationship between the complex structure of any locally isometric immersed $${\mathbb{C}Q^2_c}$$ and the complex structure of the ambient space $${\mathbb{C}P^3}$$ .