# Bott-Chern cohomology for singular complex spaces

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.

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.

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 $$H^{1,1}_{BC}(X):=H^1(X,\mathfrak R\mathcal O_X)$$.

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?

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?
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.
• If an element of $\mathfrak R\mathcal O_X$ cannot determine both real and imaginary part, why element of $\mathcal A^0/\mathfrak R\mathcal O_X$ can be described as a (1,1)-form? Sep 9 at 23:14
• If $dd^cf = 0$, then does $Re(f)$ determine $Im(f)$ up to a constant? Sep 10 at 16:30
• up to a constant, yes. Globally there could be a cohomological obstruction, think of $z\to \Re\log z$ Sep 12 at 12:18