Let $(M,I,\omega)$ be a non-compact Kähler manifold. What is the relationship between the following two properties?
- $\omega$ is $\bar{\partial}$-exact.
- $\omega$ is $d$-exact.
Are they equivalent, or does one imply the other?
2026-03-25 20:35:07.1774470907
$\bar{\partial}$-exact vs $d$-exact Kähler form
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In Hideaki Kazama and Shigeharu Takayama's book Recent Developments in Complex Analysis and Computer Algebra Chapter 13 $\partial \bar\partial$-lemma on Noncompact and Kähler Manifolds.
They prove $\partial \bar\partial$ theorem holds for toroidal groups of cohomologically finite type and that it does not hold for toroidal groups of non-Hausdorff type,that is:
You can read more details in that chapter.