Complex analytic variety

A cone is not a complex manifold, but it is a complex analytic variety.

In mathematics, particularly differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold that allows the presence of singularities. Complex analytic varieties are locally ringed spaces that are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions.

Complex analytic varieties are analogous to algebraic varieties. Roughly speaking, a complex analytic variety is a zero locus of a set of a complex analytic function, while an algebraic variety is a zero locus of a set of a polynomial function.

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