$dd^\mathbb{C}$-lemma on pair $(X,D)$

Question

Let $X$ be a Kähler manifold with a simple normal crossing divisor $D$, i.e., pair $(X,D)$. Let $\omega$ and $\omega'$ be two Kähler forms in the same Kähler class then have we $dd^\mathbb{C}$-lemma on pair $(X,D)$?

Answer 1

If you take away $D$, then $X\setminus D$ is a non-compact complex manifold, so $\partial\bar{\partial}$-lemma in general does not hold in this case. However, by the work of Bott-Chern in 1965, for any complex manifold $M$, the $\partial\bar{\partial}$-lemma always holds for characteristic forms. For example, let $(E,h)\rightarrow M$ be a hermitian vector bundle and $c_k(E,h)$ be its canonical Chern form with respect to the hermitian metric $h$, then for any other Chern form $c_k(E,h')$, we have

$c_k(E,h)-c_k(E,h')=\partial\bar{\partial}\eta$

for some $\eta\in\Omega^{2k-2}(M)$. This is called double transgression in their paper and is the key point to generalize Nevanlinna's first main theorem to higher dimensions.

I think this partially answers your question. suppose $L\rightarrow X$ is the line bundle defined by $D\subset X$, then $L|(X\setminus D)\rightarrow X\setminus D$ can be equipped with a hermitian metric $h$, and the $\partial\bar{\partial}$-lemma holds for $c_1(L|X\setminus D,h)$. This can be regarded as the $\partial\bar{\partial}$-lemma for the paring $(X,D)$.

Addendum. Given the hermitian line bundle $(L,h)$ over $X$, let $s$ be a holomorphic section of $L$ with zero set $D$, the Poincare-Lelong formula can be written as

$dd^\mathbb{C}(\log||s||^2)=-c_1(L,h)+[D]$,

the equation holds in the sense of currents. Note that this is simply a generalization of the expression of the first Chern class using a nowhere vanishing section.

Answer 2

We have $dd^c$ lemma for pair $(X,D)$, here we assume divisor $D$ is simple normal crossing and has conical singularities. In fact if two forms $\omega$ and $\omega'$ be in the same class then $\omega=\omega'+\sqrt{-1}\partial\bar\partial\varphi^D$ where $\varphi^D$ is related to Donaldson's linear theory see Simon's paper