# Complex differential geometry, complex algebraic geometry, and complex analytic geometry [closed]

From my limited understanding, complex differential geometry studies the differential geometry of complex manifolds, and complex algebraic geometry studies algebraic geometry where the underlying field is the field of complex numbers. I have come across the term complex analytic geometry in a number of places, and am puzzled as to exactly what study this refers to. What are the areas of geometry studied under complex analytic geometry? What are the relations between complex differential geometry, complex algebraic geometry, and complex analytic geometry?

## closed as too broad by user99914, Claude Leibovici, HK Lee, Frits Veerman, kingW3May 11 '17 at 10:34

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• it depends a bit on which definitions one uses, but an analytic space is something which is locally the zero-locus of several holomorphic functions on $\Bbb C^k$, while a manifold is locally $\Bbb C^n$, i.e. an analytic space may have singularities. – user8268 May 4 '17 at 20:01