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Let $X$ be a complex manifold, then we have the Poincaré lemma (or say, Dolbeault-Grothendieck lemma) (locally) on $X$, whose formulation is as follows:

( $\bar{\partial}$-Poincaré lemma) If $\alpha \in \mathcal{A}^{p, q}(B)$ is $\bar{\partial}$-closed and $q>0$, then there exists $\beta \in \mathcal{A}^{p, q-1}(B)$ with $\alpha=\bar\partial \beta$ ($B$ is the unit polydisc in $\mathbb{C}^n$).

( $\partial\bar{\partial}$-Poincaré lemma) Let $B \subset \mathbb{C}^n$ be a polydisc and let $\alpha \in \mathcal{A}^{p, q}(B)$ be a $d$-closed form with $p, q \geq 1$. Show that there exists a form $\gamma \in \mathcal{A}^{p-1, q-1}(B)$ such that $\partial \bar{\partial} \gamma=\alpha$.

$d$-Poincaré lemma is similarly as the two above.

Question: When $X$ is a complex analytic space, can we still have the similar Poincaré lemma locally on $X$?

Clues: In the past, I think one can easily get the Poincaré lemma on a complex analytic space, since we can always do the things in local model which is analytic subset of domain in $\mathbb{C}^n$. But I find a paper—Andersson, Samuelsson, A Dolbeault–Grothendieck lemma on complex spaces via Koppelman formulas (which is published on Inventiones mathematicae 2012.)—which gave the $\bar{\partial}$-Poincaré lemma on reduced complex space. So I am confused on such things.

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    $\begingroup$ On a singular complex analytic space the answer to your question heavily depends on what you mean by "differential form". The most basic definition of differential forms, namely those where you divide out the exterior derivative of the defining ideal from the ambient sheaf of forms in a local model, does not always have an exact de Rham/Dolbeault sequence. $\endgroup$
    – Thomas Kurbach
    Commented Jun 20 at 16:34
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    $\begingroup$ @LSpice Yes, you are right. $\endgroup$
    – Lelong Wang
    Commented Jun 21 at 1:35
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    $\begingroup$ @LelongWang If you can read german, then a good place to start is "Das Lemma von Poincare für holomorphe Differentialformen" by H. J. Reiffen. Thinking about it most of the reference I know for such discussions are written in german. For a non-exact Dolbeault sequence consider $f=z^4+z^2w^3+w^5$ and let $M\subset \mathbb{C}^2$ be defined by the ideal $(\partial_z f,\partial_w f)$. Then $\bar{f}$ is $\bar{\partial}$-closed on $M$ but not holomorphic of course. Assuming differential forms are defined in the basic way mentioned previously. $\endgroup$
    – Thomas Kurbach
    Commented Jun 21 at 11:00
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    $\begingroup$ The example I gave can of course also be used to see that the holomorphic de Rham sequence of $M$ is not exact. The problem here being that $M$ is not reduced. The paper "De Rham cohomology of an analytic space" by Bloom & Herrera contains a small section regarding the Poincare Lemma. They also provide a real analytic example of non-exact closed forms that if I remember correctly can also be done complex analytically, without changing the argument. $\endgroup$
    – Thomas Kurbach
    Commented Jun 21 at 11:02
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    $\begingroup$ Adding a bit on Thomas answer, what Andersson-Samuelsson prove is that the $\bar\partial$-Poincaré lemma holds for a class of sheaves that they denote $\mathcal{A}$. These sheaves coincide with the smooth forms on the regular part of $X$, and contain the smooth forms (using one definition of smooth forms) on the singular part, but the sheaves $\mathcal{A}$ might in general (have to) be strictly larger than just smooth forms. However, the definition is rather implicit, in terms of the currents you obtain when you start with smooth forms, and then apply iteratively certain integral operators. $\endgroup$
    – Richard Lärkäng
    Commented Jun 24 at 9:43

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