Understanding why nonsingular complex algebraic varieties are analytic manifolds.

Question

I am trying to understand why every (quasi-projective) nonsingular complex algebraic variety is an analytic manifold.

Consider a nonsingular affine algebraic variety $X\subset \mathbb{C}^n$ of dimension $n-k$. The idea, I think, is to write it as the level set of some holomorphic submersion. If $X$ is a complete intersection, i.e. the ideal $I(X) \subset \mathbb{C}[x_1,\dots,x_n]$ is generated by $k$ polynomials $f_1,\dots,f_k$, then using nonsingularity we have that these polynomials become the components of a submersion $f:\mathbb{C}^n\to\mathbb{C}^{k}$ and hence $X$ is the level set of a submersion, so an analytic manifold of dimension $n-k$.

The problem is that general nonsingular affine algebraic varieties $X$ are not complete intersections. However there is a theorem in Hartshorne which says that they are "locally a complete intersection".

Now I am only beginning to learn algebraic geometry and the definition of "locally a complete intersection" is in the language of schemes which I haven't learned yet. In particular I don't understand what it means geometrically.

Can we use the "locally a complete intersection" property to write $X$ as locally the level set of a submersion?

TLDR: Can we use the fact that nonsingular affine algebraic varieties are "locally a complete intersection" to write them as locally the level set of a submersion?

Answer 1

This question wasn't stated clearly enough so I asked another more precise question whose answer answers this question. See What does it mean geometrically for a variety to be locally a complete intersection?

The answer is yes, we can use the property of being a complete intersection to write a variety as locally the level set of a submersion.

Answer 2

The general result is that a smooth complex algebraic variety naturally yields a complex manifold. In more detail, Serre constructed the analytification functor from complex algebraic varieties to complex analytic spaces (see his paper GAGA). Essentially, this functor converts the Zariski topology of a variety into a complex-analytic topology. Also, it sends smooth varieties to complex manifolds.

This is the most rigorous and systematic approach to passing between algebraic and differential geometry of which I am aware.

Answer 3

This follows from the complex-analytic version of the implicit function theorem. Nonsingularity of the variety is equivalent to nonvanishing of the Jacobian, which is precisely the condition required to apply the theorem.