Biholomorphism group

Definition

For a complex manifold $X$ , its biholomorphism group or holomorphic automorphism group is

\operatorname{Aut}_{\mathrm{hol}}(X) =\{f:X\to X:f\text{ is a biholomorphism}\}.

Composition is the group operation. This is precisely the automorphism group of $X$ in the category of complex manifolds, whose morphisms are holomorphic maps.

Every biholomorphism is a diffeomorphism of the underlying smooth manifold, giving an injective homomorphism

\operatorname{Aut}_{\mathrm{hol}}(X)\hookrightarrow\operatorname{Diff}(X).

It is generally a proper subgroup because a smooth diffeomorphism need not have complex-linear differential.

Examples

The holomorphic automorphism group of $\mathbb C$ consists of affine maps $z\mapsto az+b$ with $a\neq0$ . The automorphism group of the Riemann sphere is the Möbius group

\operatorname{Aut}_{\mathrm{hol}}(\mathbb P^1(\mathbb C)) \cong PGL_2(\mathbb C).

For $\mathbb C^n$ with $n>1$ , the group includes many nonlinear automorphisms and is much larger than the complex-affine group.

Topological and Lie-group structure

The notation above first denotes an abstract group. One may equip it with a compact-open or finer topology, and under hypotheses such as compactness of $X$ , automorphism theorems give it a finite-dimensional complex Lie-group structure. Such additional structure is not automatic for an arbitrary noncompact complex manifold and is not built into the definition.

If $X$ carries a Hermitian or Kähler metric, the subgroup of holomorphic automorphisms that also preserve the metric is smaller. Its elements are the diffeomorphic instances of holomorphic isometric immersions.

References

Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: Chapters 1–2, holomorphic maps and automorphisms.
Shoshichi Kobayashi, Transformation Groups in Differential Geometry, Springer, 1972. DOI record. Relevant: automorphism groups of geometric structures.