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Let $G$ be an even-dimensional compact Lie group with Lie algebra $\mathfrak{g}$ and let $T \subset G$ be a maximal torus with Lie algebra $\mathfrak{t}$.

We can construct a left-invariant complex structure over $G$ in the following way. Let $\Sigma$ be the set of roots associated to the Cartan algebra $\mathbb C \mathfrak{t}$ (the complexification of $\mathfrak{t}$) and $\mathfrak g_\alpha$ the eigenspaces associated to the root $\alpha \in \Sigma$. We denote by $\Sigma_+$ a choice of positive roots of $\Sigma$.

Now, let $\mathfrak{m} \subset \mathbb C \mathfrak{t}$ be any subalgebra such that $\mathfrak{m} \oplus \overline{\mathfrak{m}} = \mathbb C \mathfrak{t}$, we then define $\mathfrak h = \mathfrak{m} \oplus \bigoplus_{\alpha \in \Sigma_+} \mathfrak g_\alpha$. The Lie algebra $\mathfrak h$ defines a complex structure on $G$ such that every left-translation of $G$ is holomorphic. And since $T$ is abelian, the structure $\mathfrak m$ is a bi-invariant complex structure of $T$ and thus $T$ is a complex Lie group.

In [1], it is said that $T \to G \to G / T$ is ``obviously'' a holomorphic principal bundle. I tried to verify this claim, I can prove that $G$ can be regarded as a smooth principal bundle with structure $T$, but I could not prove that this bundles is holomorphically locally trivializable.

How can I prove that such bundle is holomorphically locally trivializable? Are there some references I could follow?

I do not know if you could call this bundle a holomorphic principal bundle without this property, the the authors of [1] need to apply the Borel spectral sequence, and in the construction of such spectral sequence, it is necessary to use a local trivialization compatible with the complex structure of the fiber and of the base space.

[1]: Vanishing theorems on Hermitian manifolds - Bogdan Alexandrov and Stefan Ivanov (https://www.sciencedirect.com/science/article/pii/S0926224501000444)

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Proposition 4.26 in Félix, Oprea, Tanré, Algebraic Models in Geometry applied to $T \to G \to G/T$ shows what you want since $G/T$ is Kähler manifold. (See also Example 4.32.)

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  • $\begingroup$ Thank you very much for your answer. Unfortunately, the mentioned proposition does not answer the question. The proposition just explain how to construct a complex structure on G, which the authors claim that could be chosen to be either left or right-invariant. Still, it is not clear why this bundle would by holomorphically locally trivializable. Also, it is not clear that (in the case of G having a left-invariant structure) why the right action by T would be holomorphic (also necessary for it to be a holomorphic bundle). $\endgroup$ – Max Reinhold Jahnke Mar 4 '20 at 16:34

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