Differential forms: algebraic curves vs Riemann surfaces

Question

I've tried to read about both Riemann surfaces and algebraic curves (over an algebraically closed field) and in both cases I found definitions of objects called differential forms (or 1-forms).

For instance, here meromorphic 1-forms on a Riemann surface are introduced as a set of expressions in the form $\omega dz$ that behave in a certain way with respect to chart changes. The same kind of definition is encountered in the book Complex Algebraic Curves, F. Kirwan.

On the other hand, books on algebraic curves such as Algebraic curves, W. Fulton introduce objects called differential 1-forms in an algebraic way (see also here for instance).

So I'm wondering:

• do both those concepts coincide or are related in the case of complex curves?
• if yes, how do we show this?
• also, which of those concepts emerged first chronologically?

Answer 1

To have any hope of this being true, we need to look at global differentials on projective curves (or compact riemann surfaces).

For example, take $C = \mathbb{C}$. Then, $\Omega^{alg}_{C}(C) = \mathbb{C}[z]dz \subsetneq \Omega^{an}_C(C)$, where the left and right hand sides denote global algebraic and analytic 1-forms respectively. The inclusion is strict since there are many entire functions which are not polynomials. (eg. $e^zdz \notin \Omega_{C}^{\text{alg}}(C)$).

If $C$ is a smooth projective curve, then this is true. In fact, we can show this holds for a smooth projective variety $X$ over $\mathbb{C}$, using GAGA. The content of the GAGA theorems is that if $X^{an}$ denotes the complex manifold over $\mathbb{C}$ obtained from $X$, then there is an operation $V \mapsto V^{an}$ which takes in an algebraic vector bundle of finite rank and outputs a complex analytic one of the same rank. Furthermore, there is a natural map $V \to V^{an}$ coming from the fact that algebraic forms are always analytic. This generalizes $\Omega_C^{alg} \subset \Omega_{C}^{an}$.

The most relevant GAGA theorem to us is the following:

The map $V \to V^{an}$ induces an isomorphism of sheaf cohomology $$H^i(X, V) \to H^i(X^{an}, V^{an})$$ for all $i \geq 0$.

Taking $i = 0$, $X = C$, and $V = \Omega^{alg}_C$ gives an affirmative answer to your first question when $C$ is compact. I won't say much about the proofs in GAGA; I just wanted to point out that for your second question, this is a special case of a very general phenomenon. (In fact, GAGA is usually stated in terms of coherent sheaves; a vast generalization of finite rank vector bundles.)

Lastly, I won't say a lot about the history since I don't know it well. The algebraic differentials here are also called Kähler Differentials, which were introduced in the 1930s. On the other hand, differentials are a much older concept, and the credit for organizing (analytic) differential forms algebraically using exterior algebra is usually given to Élie Cartan and his 1899 paper.