How can we use the implicit function theorem in order to shot that a smooth projective curve is a Riemann surface ? By smoot projective curve I mean the zero locus $X$ in $\mathbb{CP}^n$ of $n-1$ homogeneous polynomials $F_1 ,...,F_{n-1}$ in $n+1$ variables such that the Jacobian matrix $\left(\frac{\partial F_i}{\partial x_i}\right)$ has maximal rank $n-1$ at every point of $X$ (I think it is also called a complete intersection curve).
Intuitively, I want to say that the implicit function theorem implies that the $x_i$ with $3\leq i \leq n+1$ are locally holomorphic functions of $x_1$ and $x_2$, with $x_1$ non zero,thus $x_1=1$ which means that we have the graph of $n-1$ holomorphic functions. But I can't find a rigorous proof of this.