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It is known that, in general, the sheaf of real analytic functions on a real analytic manifold is not coherent. However, there are some examples, where we have coherence: for example, if $X$ is a complex analytic manifold, then the sheaf of real analytic functions $A^{\omega}_X$ is the restriction to the diagonal of the sheaf $\mathcal O_{X \times \bar X}$ of complex analytic functions on the product of $X$ and its complex conjugate $\bar X$, so $A^{\omega}_X$ is coherent.

Are there other examples of real analytic manifolds for which we can prove the coherence of the structure sheaf?

I am especially interested in the case where $X$ has the metric with special holonomy, for example, when it is a $G_2$ manifold.

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  • $\begingroup$ Another approach to proving coherence is just following Oka's proof using Weierstrass division theorem for real analytic functions.This approach can be found in Houzel's expose in Seminaire Henri Cartan vol 13 no 2 (1960-1961) expose no 18 . $\endgroup$ Nov 7, 2016 at 19:17

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For real-analytic manifolds, coherence of the structure sheaf always holds. The 1-sentence reason is that one can pass to real and imaginary parts on Oka's coherence theorem in several complex variables.

Before discussing a proper proof of that 1-sentence executive summary, I should address that in the real-analytic setting however an analogue of one of Oka's "coherence" results over $\mathbf{C}$ can fail (leading one to see phrases in the literature such as "non-coherent real-analytic spaces"): there is no real counterpart to the "analytic Nullstellensatz" of Oka. To explain this, recall Oka's result that for a complex-analytic set $X$ in a complex manifold $V$ (i.e., a closed subset that is locally on $V$ given by the vanishing of finitely many holomorphic functions) the ideal sheaf of sections of $O_V$ vanishing on $X$ is always locally finitely generated (or equivalently a coherent $O_V$-module, since $O_V$ is coherent by Oka's big theorem). But this fails in the real-analytic case. In contrast, there exist real-analytic sets $Z$ in real-analytic manifolds $U$ (i.e., a closed subset that is locally on $U$ given by the vanishing of finitely many real-analytic functions) such that the ideal sheaf $I_Z$ of sections of $O_U$ vanishing on $Z$ is not locally finitely generated; such a $Z$ is called "non-coherent" in $U$ (because in such cases the subsheaf $A = O_U/I_Z$ of the sheaf of $\mathbf{R}$-valued continuous functions on $Z$ can fail to be a coherent sheaf of rings, though I have never cared enough to check up on a proof of this consequence in some cases). An explicit example of $Z$ and $U$ with $I_Z$ not coherent inside $O_U$ is given near the start of https://arxiv.org/pdf/math/0612829.pdf.

To be more detailed about the proof of coherence of the structure sheaf of a real-analytic manifold, the assertion is of local nature and so only depends on the (local) dimension; i.e., for manifolds of (pure) dimension $n$ it is equivalent to showing that the sheaf $O_U$ of real-analytic functions on every small open ball $U$ around the origin in $\mathbf{R}^n$ is coherent. By definition, this coherence amounts to local finite generation for the kernel of any $O_U$-linear map $\varphi:O_U^{\oplus N} \rightarrow O_U$ for any small $U$. This map locally "extends" over an open in $\mathbf{C}^n$; i.e., by working locally on $U$ we can arrange that that exists an open $V \subset \mathbf{C}^n$ satisfying $V \cap \mathbf{R}^n = U$ and holomorphic $F_1, \dots, F_N$ on $V$ whose restriction to $U$ recovers the $N$ components $f_1, \dots, f_N$ of $\varphi$.

By Oka's big coherence theorem (not the analytic Nullstellensatz above), the resulting map $\Phi: O_V^{\oplus N} \rightarrow O_V$ extending $\varphi$ has kernel that is locally finitely generated. Working locally on $V$ around points of $U$, we can thereby arrange that there exist $s_1, \dots, s_r \in O(V)^{\oplus N}$ generating $\ker \Phi$. We claim that the real and imaginary parts of the restrictions to $U$ of these $N$-tuples belong to $\ker \varphi$ and generate it.

Our problem is local on $V$ near $U$, so for generation we can focus on $(g_1,\dots,g_N) \in (\ker \varphi)(U)$. By working locally on $V$ around a point in $U$ we can arrange that $V$ is connected and that each $g_j$ extends (necessarily uniquely) to a holomorphic $G_j$ on $V$. Then the global section $\sum G_j F_j \in O(V)$ has restriction to $U$ equal to $\sum g_j f_j = \varphi(g_1,\dots,g_N)=0$, so $\Phi(G_1,\dots,G_N)=\sum G_j F_j = 0$ by connectedness of $V$ and analyticity considerations. Thus, working locally on $V$ some more we can arrange that $(G_1,\dots,G_N) = \sum_{k=1}^r a_k s_k$ for some $a_{1}, \dots, a_{r} \in O(V)$, so restricting to $U$ gives $$(g_1,\dots,g_N) = \sum_{k=1}^r a_{k}|_U \cdot s_k|_U.$$

Each function $g_j$ is $\mathbf{R}$-valued, whereas $a_{k}|_U$ and the $N$ components of $s_k|_U$ are $\mathbf{C}$-valued, and the real and imaginary parts of these various $\mathbf{C}$-valued functions are real-analytic on $U$. Thus, $$(g_1,\dots,g_N) = \sum_{k=1}^r ({\rm{Re}}(a_{k}|_U){\rm{Re}}(s_k|_U) - {\rm{Im}}(a_{jk}|_U) {\rm{Im}}(s_k|_U)).$$ Finally, the condition $s_k \in (\ker \Phi)(V) \subset O(V)^{\oplus N}$ says that the $N$ components $s_{k1},\dots,s_{kN} \in O(V)$ satisfy $\sum F_j s_{kj} = 0$, so $\sum f_j \cdot s_{kj}|_U = 0$ as $\mathbf{C}$-valued functions on $U$. But each $f_j$ is $\mathbf{R}$-valued, so $\sum f_j {\rm{Re}}(s_{kj}|_U)$ and $\sum f_j {\rm{Im}}(s_{kj}|_U)$ both vanish on $U$; i.e., ${\rm{Re}}(s_k|_U)$ and ${\rm{Im}}(s_k|_U)$ belong to $(\ker \varphi)(U)$. Thus, ${\rm{Re}}(s_1|_U), \dots, {\rm{Re}}(s_r|_U), {\rm{Im}}(s_1|_U), \dots, {\rm{Im}}(s_r|_U)$ generate $\ker \varphi$ over the (now shrunken) $U$. This shows that in the initial setup (before we began shrinking $U$) the kernel of $\varphi$ is locally finitely generated.

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  • $\begingroup$ Sorry for my confusion; this is more or less out of my ken. But reading on the surface, I'm having trouble reconciling the first sentence of the first paragraph with the opening of the second paragraph. (I am not doubting the authority or expertise with which the answer is written.) Does coherence hold, or does it fail? $\endgroup$
    – Todd Trimble
    Nov 7, 2016 at 12:59
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    $\begingroup$ @ToddTrimble I am not an expert either, but I think that the difference is between real analytic manifolds (that is, locally like $\mathbb{R}^n$) and real analytic sets (that is locally like the zero locus of finitely many real analytic functions). $\endgroup$ Nov 7, 2016 at 13:37
  • $\begingroup$ @DenisNardin Ah, thank you Denis. On closer reading, that does indeed seem to be what's going on. $\endgroup$
    – Todd Trimble
    Nov 7, 2016 at 14:07
  • $\begingroup$ @ToddTrimble: I have done some rewriting to clear up the points that you mention (so now your comments refer to text that no longer exists; sorry!). $\endgroup$
    – nfdc23
    Nov 7, 2016 at 15:39
  • $\begingroup$ No need for "sorry" -- thanks very much! $\endgroup$
    – Todd Trimble
    Nov 7, 2016 at 15:49

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