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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?

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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.

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  • $\begingroup$ 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? $\endgroup$
    – Hydrogen
    Sep 9 at 23:14
  • $\begingroup$ If $dd^cf = 0$, then does $Re(f)$ determine $Im(f)$ up to a constant? $\endgroup$
    – Hydrogen
    Sep 10 at 16:30
  • $\begingroup$ up to a constant, yes. Globally there could be a cohomological obstruction, think of $z\to \Re\log z$ $\endgroup$ Sep 12 at 12:18

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