I am running into conceptual problems trying to understand complex Lie groups, and more specifically, the Lie group of automorphisms $\text{Aut}(X)$ of a complex manifold $X$. Already in this question I was pointed out that certain topological restrictions are needed in order for $\text{Aut}(X)$ to be a complex Lie group.
In this sense, Kobayashi's Transformation Groups in Differential Geometry shows that for a compact complex manifold, $\text{Aut}(X)$ is a complex Lie group, whose Lie algebra corresponds to the holomorphic vector fields on $X$.
The following is a list of concepts that are not very detailed in the book, because it is assumed that the reader is already familiar with them.
Concepts I want to learn about
- How to calculate the complex Lie algebra?: In the real $C^\infty$ case, we can calculate the Lie algebra of a Lie group "by taking derivatives". For example, if we can represent the relevant Lie group as matrices, taking derivatives near the identity matrix with the relevant restrictions, e.g. $\det(A(t))=1$ for $\text{SL}(n,\mathbb{R})$, gives you elements of the Lie group. Here, real 1-parameter subgroups are needed. How does one calculate similar Lie algebra elements for complex Lie groups? Do I need $\mathbb{C}^*$ 1-parameter subgroups à la Geometric Invariant Theory? Or should I parametrize w.r.t. a real parameter $t$?
- Do complex Lie algebra elements correspond to left-invariant holomorphic vector fields as in the real Lie group theory? Left-invariance should still hold "by forgetting the holomorphic structure", I suppose.
- I assume, with no actual grounds for stating this, that the real (by forgetting holomorphic structure on $G$) and complex Lie algebra are related via complexification. This is true at the level of tangent space at the identity: does it hold for the Lie bracket as well?
- Compact real Lie groups are very important and handy: invariant measures (Haar), metrics, and Cartan 1-forms can be constructed. However, the only compact complex Lie groups are torii. Are there any generalizations of the above tools to non-Abelian complex Lie groups? I suppose some may arise via complexification of compact real groups.
Specific computations I want to be able to perform
As an illustration of Point 1 above, say we have a $\text{PGL}(2,\mathbb{C})=Aut(\mathbb{CP}^1)$, a complex Lie group of automorphisms of complex dimension 3. We know then that the action of this complex Lie group on the projective line gives a representation of Lie algebras $$ \text{Lie}(PGL(2,\mathbb{C}))\rightarrow H^0(\mathbb{CP}^1, T^{1,0}\mathbb{CP}^1). $$
Consider the subgroup of diagonal automorphisms $$ \begin{pmatrix} 1 & 0\\ 0 &\lambda \end{pmatrix} $$ for $\lambda\in\mathbb{C}^*$. I want to be able to conclude in some rigorous way that the vector field $z \frac{\partial}{\partial z}$ is the Lie algebra element generated by the above elements, where $z=Z^0/Z^1$ is the homogeneous coordinate in the complex projective line.
Should I take derivative w.r.t. $t$ of $\lambda(t) = 1 + t$? Am I able to change direction of the 1-parameter subgroup $\lambda_w(t) = 1 + w t$ for some $w\in \mathbb{C}^*$?
Any feedback on misinterpretations, conceptual errors and references to complement Kobayashi's book are more than welcome!