Category of complex manifolds

Definition

The category of complex manifolds, denoted here by $\mathbf{CMan}$ , has finite-dimensional Hausdorff second-countable complex manifolds as objects and holomorphic maps as morphisms. Identity maps and composites of holomorphic maps are holomorphic, so these data form a category. This is the house convention; disconnected objects are allowed when their component dimensions are globally bounded.

An isomorphism in $\mathbf{CMan}$ is exactly a biholomorphism: a holomorphic map with a holomorphic inverse. Merely requiring a map to be a bijection is not enough in arbitrary geometric categories; the inverse must also be holomorphic.

Relation to smooth manifolds

Forgetting the complex charts gives a functor

U:\mathbf{CMan}\longrightarrow\mathbf{Man}.

Here $\mathbf{Man}$ is the category of smooth manifolds. The functor sends a complex $n$ -manifold to its underlying real $2n$ -manifold and a holomorphic map to its underlying smooth map. This functor is faithful but not full: most smooth maps between underlying manifolds are not holomorphic. It is also not essentially surjective, because not every even-dimensional smooth manifold admits a complex structure.

Products and categorical conventions

The product $X\times Y$ , with product complex charts, is a categorical product, and a point is terminal.

The maximal subgroupoid of $\mathbf{CMan}$ retains all complex manifolds but only biholomorphisms. It should not be confused with $\mathbf{CMan}$ itself, which contains noninvertible holomorphic maps such as constant maps and branched maps.

References

Otto Forster, Lectures on Riemann Surfaces, Springer, 1981. DOI record. Relevant: holomorphic maps, biholomorphisms, and products of complex manifolds.
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, Vol. II, Wiley, 1969. Relevant: Chapter IX, complex manifolds and holomorphic mappings.