A closed analytic subset of a complex manifold is nonsingular when every point has a neighbourhood on which the subset is the zero set of a holomorphic map of full rank along that zero set. The holomorphic implicit function theorem then makes it a local coordinate subspace. This includes a closed complex submanifold, whereas an arbitrary open piece of a complex submanifold need not be globally closed in its ambient manifold.

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  1. Complex submanifold
  2. Complex manifold
  3. Integrable almost complex structure
  4. Almost complex manifold
  5. Complex structure
  6. Complex geometry
  7. Geometry and topology
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