I'm reading the book 'An Introduction to the Kahler-Ricci Flow' (Lecture Notes in Mathematics 2086). They discuss Bott-Chern cohomology on complex spaces:


Let $X$ be a complex space(i.e. analytic variety) with normal singularity.



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> Lemma 4.6.1. Any pluriharmonic distribution on $X$ is locally the real part of a holomorphic function, i.e. the kernel of the $dd^c$ operator on the sheaf $\mathcal D'\_X$ of germs of distributions coincides with the sheaf $\mathfrak R\mathcal O\_X$ of real parts of holomorphic germs.
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> Definition 4.6.2. A (1,1)-form (resp.(1,1)-current) with local potentials on $X$ is defined to be a section of the quotient sheaf $\mathcal C^\infty\_X/\mathfrak R\mathcal O\_X$(resp. $\mathcal D'\_X/\mathfrak R\mathcal O\_X$). We also introduce the Bott-Chern cohomology space $H^{1,1}\_{BC}(X):=H^1(X,\mathfrak R\mathcal O\_X)$.
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My questions are:


1. The lemma seems suspicious to me. It implies if $dd^cf = 0$ for a smooth function $f$, then $f = Re(h)$ for some holomorphic function $h$. So, $f$ must be a real function? At least $if$ also satisfies the condition which is not real.
2. They say a (1,1)-form with local potentials can be described as a closed $(1,1)$-form $\theta$ on $X$ that is locally of the form $\theta = dd^cu$ for a smooth function $u$. I wnt to prove this. First, I need to use the short exact sequence
$$0\rightarrow\mathfrak R\mathcal O\_X\rightarrow \mathcal A^0\_X\rightarrow \mathcal A\_X^0/\mathfrak R\mathcal O\_X\rightarrow 0$$
but I have no idea explicitly what the first map is?
3. They mention that a closed $(1,1)$-forms and currents on $X$ are not necessary locally $dd^c$-exact in general. But in my knowledge(I may be wrong), it holds when $X$ is a manifold. What makes it different when $X$ is singular?



Bott-Chern cohomology for singular complex spaces

> 
> closed (1,1)-forms and currents on X
> are not necessary locally $dd^c$-exact in general
> What makes it different when X is singular?
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The obstruction to local $dd^c$-lemma
is $R^1\pi\_\*(O\_{X'})$, where
$\pi:\; X' \to X$ is the resolution of
singularities. When $X$ is smooth
(more generally, when it
has rational singularities), this sheaf is
trivial, but if the singularities are bad
(say, when $X$ is the cone over a curve
of big genus), local $dd^c$-lemma might
fail.



> 
> It implies if $dd^cf=0$ for a smooth function $f$, then
> $f=Re(h)$ for some holomorphic function $h$. So, $f$ must be a
> real function?
> 
> 
> 


This is usually assumed. If $f$ is complex,
its real and imaginary parts are both real
parts of (a priori, different) holomorphic
functions.


