Complex analytic variety

Summary

A complex analytic variety, or complex analytic set, in a complex manifold $X$ is a closed subset with a local description. Near each point of $X$ it is the common zero set of finitely many holomorphic functions. It is the analytic counterpart of an algebraic variety, with polynomials replaced by holomorphic functions. Analytic varieties may have singular points, and their local structure is studied through germs of analytic sets and the ring of convergent power series. Abstract varieties of this kind, defined by gluing local models, are the complex analytic spaces.