# Holomorphic local trivialization of a principal toric bundle

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)

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.)