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?
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Complex analytic geometry is what we get when we transfer all the definitions from algebraic geometry into the holomorphic category. That is, instead of an algebraic variety being defined locally by the zero set of polynomials, an analytic variety is defined locally by the zero set of holomorphic functions.
As there are more holomorphic functions than polynomials, and holomorphic or meromorphic functions can behave much more erratically than polynomials or rational functions, carrying out the standard constructions of algebraic geometry is now much more difficult than in the algebraic category. For example, the statement that the direct image of a coherent sheaf under a proper map is coherent is a major theorem involving heavy functional analysis in analytic geometry.
Generally, life in analytic geometry is just more difficult. The exception is when we deal with smooth, compact manifolds, where we can use the tools of differential geometry to tame analytic difficulties, and get results that often go beyond what is easily provable by purely algebraic methods. The cost of these results is that it's usually totally non-obvious what to do when our spaces are no longer smooth.
The relations between complex algebraic geometry and complex analytic geometry have intensively studied by Serre in the GAGA.