Gaussian curvature of a complex projective curve

Question

Let $X \subset \mathbb CP^2$ be a complex curve inheriting metric from $\mathbb CP^2$. Suppose that locally $X$ is given by a holomorphic map $z \to [h_1(z) \colon h_2(z) \colon h_3(z)]$. What is the easiest way to compute the Gaussian curvature of $X$ at a fixed point? I was thinking about some analog of the Gauss map allowing to express the curvature in terms of the determinant of its differential but I haven't found any good reference. The second approach that I see is to compute the second fundamental form of $X$ but I don't know if there is some explicit formula for it in the described case. Please help me with a correct direction of thinking.