Complex submanifold - Imported Knowlpedia Content

Definition

Let $X$ be a complex manifold of complex dimension $n$ . A subset $Y\subseteq X$ is a complex submanifold of complex dimension $k$ if every $p\in Y$ has a complex chart $(U,\varphi)$ of $X$ , with $\varphi(p)=0$ , such that

\varphi(U\cap Y)=\varphi(U)\cap(\mathbb C^k\times\{0\}) .

The induced charts make $Y$ a complex manifold, and its inclusion into $X$ is holomorphic. In particular, $Y$ is an embedded submanifold of the underlying smooth manifold, of real dimension $2k$ and real codimension $2(n-k)$ .

Equivalent formulation

An injective holomorphic immersion $i:Y\to X$ whose underlying map is a topological embedding identifies $Y$ biholomorphically with a complex submanifold of $X$ . Conversely, every complex submanifold inclusion has these properties. The local coordinate form follows from the holomorphic constant-rank theorem Wells, Chapter I, §3.

Tangent spaces and local equations

For $p\in Y$ , the inclusion identifies the tangent space $T_pY$ with a complex-linear subspace of $T_pX$ . Locally, a codimension- $r$ complex submanifold is the common zero set of $r$ holomorphic functions whose complex differentials are linearly independent along that zero set.

Examples and non-examples

A complex linear subspace of $\mathbb C^n$ and the graph of a holomorphic map are complex submanifolds. The real line $\mathbb R\subset\mathbb C$ is a smooth embedded submanifold but not a complex submanifold: its real tangent line is not preserved by multiplication by $i$ .

Conventions and scope

A complex analytic subset may have singular points and therefore need not be a complex submanifold. “Complex analytic submanifold” in the alias list refers only to the nonsingular, locally coordinate-linear notion defined here.

References

R. O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. Springer DOI record. Relevant: Chapter I, §3, holomorphic maps and submanifolds.
D. Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Springer DOI record. Relevant: §2.1, complex manifolds and holomorphic submanifolds.