Complex Manifolds

I've drafted an initial article on Complex Manifolds. Please review.

Review by Critic Carla:

The article is technically sound and well-structured, but it contains a significant factual error regarding a key historical figure and lacks a few essential conceptual bridges.

• Correct factual error in the "Applications" section: The text states that the GAGA principle was "formulated by Jean GAGA (Jean-Pierre Serre)." GAGA is an acronym (Géométrie Algébrique et Géométrie Analytique), not a person's name. This should be corrected to state that the principle was formulated by Jean-Pierre Serre.
• Expand on the "Almost Complex Structure" section: While the article mentions the Nijenhuis tensor, it would be more informative to explicitly state the Newlander-Nirenberg Theorem, which formally provides the condition for an almost complex structure to be integrable.
• Add a section on Stein Manifolds: The article covers compact manifolds and projective varieties well, but it misses "Stein manifolds," which are the complex analogues of affine space and are fundamental to the study of non-compact complex manifolds.
• Clarify the "Notable Contributors" in the infobox: The entry "Sheaves and cohomology theorists" is too vague for an encyclopedia. Replace this with specific names of influential mathematicians (e.g., Cartan, Oka, or Kodaira) who developed these tools.

Fact-check by Fact-Checker Finn:

• Factual Error: The article states the GAGA principle was formulated by "Jean GAGA (Jean-Pierre Serre)." GAGA is an acronym for Géométrie Algébrique et Géométrie Analytique; it is not a person's name.
• Vague/Imprecise Phrasing: The text refers to "mathematicians such as sheaves and cohomology theorists." Sheaves and cohomology are mathematical tools/theories, not names or categories of people.
• Missing Nuance: The claim that "any holomorphic function on a compact connected complex manifold must be constant" is correct, but it should be clarified that this applies specifically to functions mapping to $\mathbb{C}$ (scalar-valued functions).
• Inaccurate Generalization: The text states that Hodge theory and the $\partial\bar{\partial}$-lemma "have no direct analogues in the purely real setting." While these specific formulations are complex-analytic, there are significant analogues in real differential geometry (e.g., De Rham cohomology and Hodge theory for Riemannian manifolds).

Review complete. Some issues were flagged. @Scientist Sage, please address the feedback.