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In complex geometry, there is the a lemma, analogous to the Poincaré lemma in (real) differential geometry, which states that a $(p,q)$-form that is $\bar\partial$-closed is locally $\bar\partial$-exact. In the book Complex Geometry by Huybrechts, we find the following remark:

The [...] proposition and its corollary are known as the Grothendieck-Poincaré lemma. The first proof of it is due to Grothendieck and was presented by Serre in the Séminaire Cartan in 1958.

The claim that this lemma was (first) proven by Grothendieck is also backed up by these notes, in section 5. Now, I would like to see the original presentation of the proof, so I tried to find the source that Huybrechts refers to. The obvious thing to do is go through the Séminaire Cartan scripts. However, in the volumes corresponding to the year 1958 there is no contribution by Serre. In fact, I don't think there is anything related to the Grothendieck-Poincaré lemma in the entire collection (I convinced myself of this by skipping through all of Serre's contributions, which can be found with a little bit of effort, e.g. on this page.

Therefore, I would like to know the following:

  • Did Serre present this proof by Grothendieck anywhere (else)? Where, and when?
  • If not, where and when was the proof due to Grothendieck first published or otherwise made public?
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  • $\begingroup$ Interesting. Note that some people also append Dolbeault to the name. I was just trying to find Dolbeault's *Sur la cohomologie des variétés analytiques complexes*—it simply must contain the statement—to check the references therein, but I can't access it. Anyone else? $\endgroup$ – Ben Sep 14 '16 at 16:32
  • $\begingroup$ I found it, and it contains no references whatsoever. Next to the statement, which he attributes to Grothendieck, Dolbeault writes Cette démonstration diffère de celle de Grothendieck et m'a été communiquée par H. Cartan (The proof is different from Grothendieck's and was communicated to me by H. Cartan.) There is no mentioning of the Séminaire, however. $\endgroup$ – Ben Sep 14 '16 at 17:34
  • $\begingroup$ @Ben yes, I'm aware that Dolbeault also "features in this story"---the book on Riemann surfaces by Forster calls the one-dimensional case the Dolbeault lemmma :) thanks for checking out that source! $\endgroup$ – Danu Sep 14 '16 at 18:37
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An alternative name is the Dolbeault-Grothendieck lemma. Dolbeault himself writes the following:

"It is proved by P. Dolbeault in the $C^\omega$ case, by homotopy, as can been [sic!] the Poincare lemma. H. Cartan brings the proof to the $C^\omega$ case by a potential theoritical method [Do 53]. Simultanously, the lemma has been proved by A. Grothendieck, by induction on the dimension, from the case $n = 1$ a consequence of the non homogeneous Cauchy formula, see [Ca 53], expose 18."

[Ca 53] are the notes of Séminaire Cartan 1953/54, expose 18 is indeed by Serre, and the result is given by Proposition 1. It is likely that 1958 was a typo. [Do 53] are two short notes in Comptes Rendus:

P. Dolbeault, Sur la cohomologie des varietes analytiques complexes, C.R. Acad. Sci. Paris 236 (1953), 175-177.

P. Dolbeault, Sur la cohomologie des varietes,analytiques complexes, II, C.R. Acad. Sci. Paris 236 (1953), 2203-2205.

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