Quoting from Huybrecht's book Complex Geometry on the $\partial\overline{\partial}$-lemma for Kaehler manifolds:

Although it looks like a rather innocent technical statement, it is crucial for many results.

Later in the book the result is used in the study of formality of Kaehler manifolds, but what other important applications does it have?


This is a list of length one :)

The $\partial \bar \partial$-Lemma allows a parameterization of the cohomology class $[\omega]$ of a compact Kaehler manifold $(M, \omega)$ by means of scalar functions, i.e. $$ [\omega] = \{ \omega + i \partial \bar \partial u: u \in C^\infty(M) \} $$

The first thing I can think of is that it allows to boil down tensorial equation to scalar equations.

An example of this is the Calabi-Yau theorem:

On a compact Kaehler manifold $(M, \omega)$, for every closed $(1,1)$ form $\rho \in 2 \pi c_1(M)$, there exist a unique Kaehler metric $\omega' \in [\omega]$ such that its Ricci form equals $\rho$.

In order to obtain a scalar PDE, one can just use the Ansatz that a solution must be of the form $\omega + i \partial \bar \partial u$ for some smooth $u$, but the $\partial \bar \partial$-Lemma is used to prove the uniqueness of the solution in the cohomology class. Namely, two solutions must differ by the $\partial \bar \partial$ of a function.

For this I can refer you to Moroianu's book (Calabi-Yau chapter) or the good old Besse.


There is a simpler version of what David has written.

Claim. Given a (1,1) form $\alpha$ representing the first Chern class of a holomorphic line bundle $[\alpha]=c_1(L)$, one can find a metric $h$ on $L$ such that its curvature is $$ \Theta(L,h)=\frac{i}{2\pi}\alpha. $$

Indeed, let $h_0$ be an arbitrary metric background on $L$ and $h=e^\phi h_0$ is the metric we are looking for. Then $$\frac{i}{2\pi}\alpha=\Theta(L,h)=\Theta(L,h_0)+\partial\bar\partial\phi.$$

Existence of such $\phi$ immediately follows from $\partial\bar\partial$-lemma, applied to $d$-exact (1,1)-form $\frac{i}{2\pi}\alpha-\Theta(L,h_0)$.

Remark. This is not true in non-Kaehler case, for example on Hopf surface $M$ canonical bundle $K_M$ is topologically trivial, while it does not admit flat metric.

The claim has important consequence: line bundle $L$ is positive if and only if its first Chern class has positive $(1,1)$ representative.

See also https://mathoverflow.net/q/241687 where essentially the same claim is discussed.


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