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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

  1. 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 $t$?
  2. 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.
  3. 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?
  4. 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!

topolosaurus
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    This is a very mature, well expressed question: bravo! By the way, I have just answered one of your previous questions. – Georges Elencwajg Apr 04 '22 at 08:58
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    A couple of remarks on concepts 3 and 4: 3 is not quite right. Forgetting the holomorphic structure $G$ is a real Lie group but its Lie algebra does not complexify to the complex Lie algebra. Instead it is the exact same Lie algebra as the original but viewed as a real Lie algebra (so it has real dimension twice as big as the original complex dimension). As to 4 while there are a lot of tools available to compact Lie theory that don't hold in the noncompact case, a lot of things in the study of Lie theory really come from looking at the complex ones. – Callum Apr 05 '22 at 12:08
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    Metrics and measures are usually a compact Lie group thing (at least if you want your metric to be positive definite) but the Cartan form (assuming you are talking about the Maurer-Cartan form) is defined for any Lie group. – Callum Apr 05 '22 at 12:12
  • Thank you for your insights! Any chance you might be willing to clarify my naive misunderstanding of how to compute complex Lie algebra elements by differentiating at the identity? – topolosaurus Apr 05 '22 at 12:52
  • $G$ is still the same manifold so it has the same tangent space. Remember a complex manifold of dimension $n$ is a real manifold of dimension $2n$. – Callum Apr 05 '22 at 13:34
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    Note this is different to finding real forms. The real form of a complex Lie algebra $\mathfrak{g}$ is indeed a real Lie algebra which complexifies to $\mathfrak{g}$. In general there are many different ways to do this (even up to isomorphism) providing different real forms. For a semisimple Lie algebra there is always at least a compact real form and a split real form but in general there are many more. Doing this for Lie groups is possible but we have to be somewhat more careful as to our definition of real form - there are real Lie groups which have no complexification in the naive sense – Callum Apr 05 '22 at 13:40
  • Ah I've just realised I misread your comment after all that. You can do either option in your question 1 (I think) to get the same answer. Obviously if you just look at real 1-parameter subgroups you will be ignoring the complex structure on $\mathfrak{g}$ but it will still get you all the elements. – Callum Apr 05 '22 at 13:54

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