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Let $M$ be a real analytic manifold. In the book "Sheaves on Manifolds" by Kashiwara and Schapira it is claimed on p. 127 (without reference or proof) that

  1. the Poincare lemma holds for the de Rham complex of real analytic differential forms,

  2. the cohomology of the sheaves of real analytic analytic differential $p$-forms vanishes in degree $\geq 1$ and for all $p \geq 0$.

Is there a reference for 1.? Why does 2. hold?

Ad 1.: As far as I see, one can prove the first claim as for the de Rham complex of smooth differential forms. More precisely, for $M$ contractible the usual homotopy to $0$ of the smooth de Rham complex restricts to a homotopy of the real analytic de Rham complex. The key point is that one can differentiate analytic functions under an integral sign and that the derivative is analytic again. But to use this result a reference would be appreciated.

Ad 2.: In an other place the authors mention Grauert's result that every real analytic manifolds $M$ can be embedded into a Stein space $X$. For Stein spaces one knows that the higher cohomology of any coherent sheaf vanishes. This might to be a starting point for proving 2. But of course this result cannot applied directly because the sheaves of differentials on $M$ will not even be modules over $\mathcal{O}_X$. Maybe one can work with real analytic forms that have complex coefficients? I am just fishing in murky waters...

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    $\begingroup$ For #1, are you asking for a reference in the real-analytic or holomorphic cases? If the former and a reference is not found one could say "as an application of the Cauchy-Riemann equations" (the real-analyticity of the usual integral is a "local" problem, so it is enough to treat holomorphicity on a polydisc for $z \mapsto \int_{0}^1 f(tz){\rm{d}}t$, to which we apply the theorem on passing partial derivatives through the integral of real and imaginary parts and the CR equations in each coordinate direction, the latter characterizing holomorphicity for smooth $\mathbf{C}$-valued functions). $\endgroup$ – user74230 Apr 17 '15 at 11:26
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    $\begingroup$ For #2, by the discussion from Theorem 5.41 to Proposition 5.42 in "Stein to Weinstein and back" by Cieliebak and Eliashberg, $M$ is a "totally real" submanifold of some such $X$, so the complexified cotangent bundle of $M$ is the pullback of the holomorphic cotangent bundle of $X$. So it suffices that ${\rm{H}}^n(M, \Omega^p_X)=0$ for $n>0$. By Prop. 5.42 above, $M$ has arbitrarily small Stein open neighborhoods $U_j$ in $X$. By Thm. 4.11.1 Ch. II in Godement's sheaf theory book, that cohomology is the direct limit of the ${\rm{H}}^n(U_j,\Omega^p_X)$'s, which vanish since each $U_j$ is Stein. $\endgroup$ – user74230 Apr 17 '15 at 13:07
  • $\begingroup$ In my comment above, I should have mentioned that $M$ is also closed in $X$ (to avoid any confusion about the intended sense of "submanifold"). $\endgroup$ – user74230 Apr 17 '15 at 15:02
  • $\begingroup$ Just shamelessly jumping on the opportunity to ask you: when will we have the joy to read the second volume of your co-authored masterful treatise ? :-) $\endgroup$ – Georges Elencwajg Apr 18 '15 at 9:25
  • $\begingroup$ @UserWithOnlyNumber: Thank you very much. $\endgroup$ – Torsten Wedhorn Apr 23 '15 at 3:31
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Regarding question 2 :This is answered in the paper of Henri Cartan Bulletin SMF vol 85 yr 1957 pages 77-99 .It is essentially as outlined by user74230 .Question 1:If you look at the proof of Poincare lemma in Narasimhan's book Analysis on Real and Complex manifolds pages 128 to 129 it is clear that the proof works for real analytic forms .We just need to observe that the coefficients are given locally by convergent power series and the operations involved in the proof preserve them.

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